Cointegration and Forward and Spot Exchange Rate Regressions

_Problem: Policy make and theory; The rational choice of individual and collective; Individual expect ional Hypothesis; the optimizing model in the real world
_Different measures(2.8, 2.9; 2.10, 2.11; 2.13,2.14; 2.14)
_ A revise one: RE = et·（I*T / IT ）·[(1n*)+n*·(I*N/ IT * )] / [(1-n)+n·(IN/ IT )]国际金融 研究\方论文修改.doc 国际金融研究\三大经 济体实际汇率图.xls
On the background of international finance On the method of studies On the References
参考文献
克鲁格曼和奥伯斯法尔德，《国际经济学》，人大出版社， 2001
Salvatore，《国际经济学》，清华，1997 Appleyard and Field， 《国际经济学》，机械工业出版社，
GDP, how much can the firm get 1.5800 * 0.9120 * 100M = 1.4410 *100M EUR. Problem : how much can the firm get JPY against 100M
EUR 121.45/0.9130 *100m = 133.13 * 100M EUO
USD (8.7250 – 3.7420)/3.7420 = 133% RMB ( 3.7420 – 8.7250)/8.7420 = 57%

5.1 Vector Autoregressions
323
5.1.2
Properties of a VAR(1)
The properties of the VAR(1) are fairly simple to study. More importantly, section 5.2 shows that all VAR(P)s can be rewritten as a VAR(1), and so the general case requires no additional effort than the first order VAR.
In many situations, analyzing a time-series in isolation is reasonable; in other cases univariate analysis may be limiting. For example, Campbell (1996) links financially interesting variables, including stock returns and the default premium, in a multivariate system that allows shocks to one variable to propagate to the others. The vector autoregression is the mechanism that is used to link multiple stationary time-series variables together. When variables contain unit roots, a different type of analysis, cointegration, is needed. This chapter covers these two topics building on many results from the analysis of univariate time-series.

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JOURNAL OF POLITICAL ECONOMY
of Blanchard and Kiyotaki (1987) and Ball and Romer (1989) in an analytically tractable, dynamic, two-country framework. Section II sets out an infinite-horizon monetary model of a monopolistically competitive world economy. We show how to solve for the long-run and short-run equilibria of a log-linearized version of the model. In Section III we analyze positive and normative aspects of monetary and fiscal policy. Section IV catalogs a number of possible extensions of the model, and Section V presents conclusions. Various elements of our approach can be found in earlier work by several authors. Each component of Mussa's (1984) aggregative model is inspired by individual maximization, but the model as a whole lacks an integrative foundation. McKibbin and Sachs (1991) and Stockman and Ohanian (1993) develop numerical sticky-price models that incorporate intertemporal maximization but lack foundations on the supply side. The model of Calvo and V6gh (1993) assumes sticky prices and demand-determined output but presents no rationale for the latter assumption. Also, its small-country setting prevents analysis of international transmission issues. Romer (1993) models a world of two interacting monopolistically competitive economies, but his analysis is static and its microfoundations are not fully specified. Dixon (1993) surveys other static open-economy models based on imperfect competition. Perhaps the closest precursor to our study is the paper by Svensson and van Wijnbergen (1989); but its assumption of perfectly pooled international risks, aside from uneasily matching its pricing and rationing assumptions, precludes discussion of the international wealth redistributions that are central to our analysis.'

Definition of cointegration
Let Yt = (y1t, . . . , ynt)’ denote an (n×1) vector of I(1) time series. Yt is cointegrated if there exists an (n×1) vector β = (β1, . . . , βn)’ such that β’ Yt = β1y1t + · · · + βnynt ∼ I(0) a linear combination of non-stationary variables that is stationary long-run equilibrium relationship variables cannot drift too far apart from the equilibrium
ADF and PP unit root tests applied to the estimated cointegrating residual do not have the usual Dickey-Fuller distributions (Phillips and Ouliaris (1990)) The distribution depend on the deterministic terms and the number of variables
y1t y1,t 1 vt y2 t y1t ut y3t y1t t
Contents
1. Spurious Regression 2. Definition of cointegration 3. Residual-Based Tests for Cointegration(EG two-step method) 4. A example: price indexes 5. Johansen cointegration test 6. A example: money quantity

外文题目：Can macroeconomic variables explain long term stock market movements?A comparison of the US and Japan出处：School of Economics and Finance, University of St Andrews, St Andrews, UK,作者：Andreas Humpe Peter MacmillanCan macroeconomic variables explain long term stock marketmovements?A comparison of the US and JapanBy Andreas Humpe Peter Macmillan原文：ABSTRACTWithin the framework of a standard discounted value model we examine whether a number of macroeconomic variables influence stock prices in the US and Japan. A cointegration analysis is applied in order to model the long term relationship between industrial production, the consumer price index, money supply, long term interest rates and stock prices in the US and Japan. For the US we find the data are consistent with a single cointegrating vector, where stock prices are positively related to industrial production and negatively related to both the consumer price index and a long term interest rate. We also find an insignificant (although positive) relationship between US stock prices and the money supply. However, for the Japanese data we find two cointegrating vectors. We find for one vector that stock prices are influenced positively by industrial production and negatively by the money supply. For the second cointegrating vector we find industrial production to be negatively influenced by the consumer price index and a long term interest rate. These contrasting results may be due to the slump in the Japanese economy during the 1990s and consequent liquidity trap.Keywords: Stock Market Indices, Cointegration, Interest Rates.I. Introduction.A significant literature now exists which investigates the relationship between stock market returns and a range of macroeconomic and financial variables, across a number of different stock markets and over a range of different time horizons. Existing financial economic theory provides a number of models that provide a framework for the study of this relationship.One way of linking macroeconomic variables and stock market returns is through arbitrage pricing theory (APT) (Ross, 1976), where multiple risk factors can explain asset returns. While early empirical papers on APT focussed on individual security returns, it may also be used in an aggregate stock market framework, where a change in a given macroeconomic variable could be seen as reflecting a change in an underlying systematic risk factor influencing future returns. Most of the empirical studies based on APT theory, linking the state of the macro economy to stock market returns, are characterised by modelling a short run relationship betweenmacroeconomic variables and the stock price in terms of first differences, assuming trend stationarity. For a selection of relevant studies see inter alia Fama (1981, 1990), Fama and French (1989), Schwert (1990), Ferson and Harvey (1991) and Black, Fraser and MacDonald (1997). In general, these papers found a significant relationship between stock market returns and changes in macroeconomic variables, such as industrial production, inflation, interest rates, the yield curve and a risk premium.An alternative, but not inconsistent, approach is the discounted cash flow or present value model (PVM)1. This model relates the stock price to future expected cash flows and the future discount rate of these cash flows. Again, all macroeconomic factors that influence future expected cash flows or the discount rate by which these cash flows are discounted should have an influence on the stock price. The advantage of the PVM model is that it can be used to focus on the long run relationship between the stock market and macroeconomic variables. Campbell and Shiller (1988) estimate the relationship between stock prices, earnings and expected dividends. They find that a long term moving average of earnings predicts dividends and the ratio of this earnings variable to current stock price is powerful in predicting stock returns over several years. They conclude that these facts make stock prices and returns much too volatile to accord with a simple present value model. Engle and Granger (1987) and Granger (1986) suggest that the validity of long term equilibria between variables can be examined using cointegration techniques. These have been applied to the long runrelationship between stock prices and macroeconomic variables in a number of studies, see inter alia Mukherjee and Naka (1995), Cheung and Ng (1998), Nasseh and Strauss (2000), McMillan (2001) and Chaudhuri and Smiles (2004). Nasseh and Strauss (2000), for example, find a significant long-run relationship between stock prices and domestic and international economic activity in France, Germany, Italy, Netherlands, Switzerland and the U.K. In particular they find large positive coefficients for industrial production and the consumer price index, and smaller but nevertheless positive coefficients on short term interest rates and business surveys of manufacturing. The only negative coefficients are found on long term interest rates. Additionally, they find that European stock markets are highly integrated with that of Germany and also find industrial production, stock prices and short term rates in Germany positively influence returns on other European stock markets (namely France, Italy, Netherlands, Switzerland and the UK).In this paper, we will draw upon theory and existing empirical work as a motivation to select a number of macroeconomic variables that we might expect to be strongly related to the real stock price. We then make use of these variables, in a cointegration model, to compare and contrast the stock markets in the US and Japan. In contrast to most other studies we explicitly use an extended sample size of most of the last half century, which covers the most severe stock market booms in US and Japan. While Japans’ h ay days have been in the late 1980s, the US stock market boom occurred during the 1990s and ended in 2000. Japans’ stock market has not yet fully recovered from a significant decline during the 1990s, and at the time of writing, trades at around a quarter of the value it saw at its peak in 1989.2The aim of this paper is to see whether the same model can explain the US and Japanese stock market while yielding consistent factor loadings. This might be highly relevant, for example, to private investors, pension funds and governments, as many long term investors base their investment in equities on the assumption that corporate cash flows should grow in line with the economy, given either a constant or slowly moving discount rate. Thus, the expected return on equities may be linked to future economic performance. A further concern might be the impact of the Japanese deflation on real equity returns. In this paper, we make use of the cointegration methodology, to investigate whether the Japanese stock market has broadly followed the same equity model that has been found to hold in the US.In the following section, we briefly outline the simple present value model of stock price formation and make use of it in order to motivate our discussion of the macroeconomic variables we include in our empirical analysis. In the third section we briefly outline the cointegration methodology, in the fourth section we discuss our results and in the fifth section we offer a summary and some tentative conclusions based on our results.As suggested by Chen, Roll and Ross (1986), the selection of relevant macroeconomic variables requires judgement and we draw upon both on existing theory and existing empirical evidence. Theory suggests, and many authors find, that corporate cash flows are related to a measure of aggregate output such as GDP or industrial production4. We follow, Chen, Roll and Ross (1986), Maysami and Koh (1998) and Mukherjee and Naka (1995) and make use of industrial production in this regard.Unanticipated inflation may directly influence real stock prices (negatively) through unexpected changes in the price level. Inflation uncertainty may also affect the discount rate thus reducing the present value of future corporate cash flows. DeFina (1991) has also argued that rising inflation initially has a negative effect on corporate income due to immediate rising costs and slowly adjusting output prices, reducing profits and therefore the share price. Contrary to the experience of the US, Japan suffered periods of deflation during the late 1990s and early part of the 21st century, and this may have had some impact on the relationship between inflation and share prices. The money supply, for example M1, is also likely to influence share prices through at least three mechanisms: First, changes in the money supply may be related to unanticipated increases in inflation and future inflation uncertainty and hence negatively related to the share price; Second, changes in the money supply may positively influence the share price through its impact on economic activity; Finally, portfolio theory suggests a positive relationship, since it relates an increase in the money supply to a portfolio shift from non-interest bearing money to financial assets，including equities. For a further discussion on the role of the money supply see Urich and Wachtel (1981), Rogalski and Vinso (1977) and Chaudhuri and Smiles (2004). The interest rate directly changes the discount rate in the valuation model and thus influences current and future values of corporate cash flows. Frequently, authors have included both a long term interest rate (e.g. a 10 year bond yield) and a short term interest rate (e.g. a 3 month T-bill rate)5. We do not use a short term rate as our aimhere is to find a long term relationship between the stock market and interest rate variables. Changes in the short rate are mainly driven by the business cycle and monetary policy. In contrast, the long term interest rate should indicate the longer term view of the economy concerning the discount rate.For our long term rate using the US data we have chosen a 10 year bond yield, however for Japan we instead use the official discount rate. This is due largely to data availability constraints and that the market for 10 year bonds (and similar maturities) in Japan has not been liquid for most of the time period covered. We should note that the discount rate is an official lending rate for banks in Japan, which implies it is generally lower than a market rate. Unlike many studies; see inter alia Mukherjee and Naka (1995) and Maysami and Koh (2000); we do not include the exchange rate as an explanatory variable. We reason the domestic economy should adjust to currency developments and thus reflect the impact of fo reign income due to firms’ exports measured in domestic currency over the medium run. Additionally, in the white paper on the Japanese Economy in 1993, published by the Japanese Economic Planning Agency, it has been pointed out that the boom in the economy during the late 1980s has been driven by domestic demand rather than exports (Government of Japan, 1993).In contrast to our study, many researchers have based their analysis on business cycle variables or stock market valuation measures such as the term spread or 7default spread for the former category or dividend yield or earnings yield for the latter. Examples of those papers include Black, Fraser and MacDonald (1997); Campbell and Hamao (1992); Chen, Roll and Ross (1986); Cochran, DeFina and Mills (1993); Fama (1990); Fama and French (1989); Harvey, Solnik and Zhou (2002) and Schwert (1990). These variables are usually found to be stationary and as we plan to model long term equilibrium using non stationary variables we do not included them in our model.As McAdam (2003) has confirmed, the US economy has been characterized by more frequent but less significant downturns relative to those suffered by the Japanese economy. This might be explained by a higher capital and export orientated Japanese economy relative to the US. We might therefore expect higher relative volatility in corporate cash flows and hence also in Japanese share prices. A priori, therefore, share prices in Japan may be more sensitive to changes in industrial production, although the greater relative volatility may also influence the estimated coefficient standard errors in any regression equation. However, previous research (seeBinswanger, 2000) has found that although both stock markets move positively with economic output, that the coefficient for output on equity returns tends to be larger for the US data relative the Japanese data. Campbell and Hamao (1992) also find smaller positive coefficients for the dividend price ratio and the long-short interest rate spread on stock market returns in Japan relative to the US in a sample covering monthly data from 1971 to 1990. Thus the intuitive expectation of higher coefficients in Japan due to higher capital and export exposure has not been confirmed empirically in existing research. Japan’s banking crisis and subsequent asset deflation during the 1990s could have changed significantly the influence of a number of variables, particularly the interest rate and money supply.To our knowledge, there has not been any empirical study of the present value model in Japan after the early 1990s and thus the severe downturn with low economic growth and deflation. Existing studies of the Japanese stock market before 1990 include Brown and Otsuki (1990), Elton and Gruber (1988), and Hamao (1988), although these papers mainly consider stationary business cycle variables and risk factors and therefore cannot give an indication of the empirical relationships we might expect to find in our model.In this paper we compare the US and Japan over the period January 1965 until June 2005. The use of monthly data gives the opportunity to analyse a very rich data set, to our knowledge earlier papers have only analysed shorter periods or have made use of a lower data frequency. This allows us to include the impact of the historically high volatility of both stock markets. The US stock market showed very high returns between 1993 and 1999, while from 2000 until 2003 returns have been very large and negative. In the Japanese stock market during the period from 1980 through to 1990, returns have in the main been large and positive while they tend to have been large negative for most of years between 1990 and 2003. The impact of recent problems to the Japanese banking sector is also captured in our data (see Government of Japan, 1993). Most existing research has been applied to US data and very little is known about the differences between US and Japanese stock market valuation. This paper investigates the differences and common patterns in both countries in order to verify whether the same variables that explain aggregate stock market movement in the US can also be used to do so in Japan.ConclusionIn order to achieve a deeper understanding of long term stock market movements, a comparison of the US and Japanese stock market, using monthly data over the last 40years has been conducted. Using US data we found evidence of a single cointegration vector between stock prices, industrial production, inflation and the long interest rate. The coefficients from the cointegrating vector, normalised on the stock price, suggested US stock prices were influenced, as expected, positively by industrial production and negatively by inflation and the long interest rate. However, we found the money supply had an insignificant influence over the stock price. In Japan, we found two cointegrating vectors. One normalised on the stock price provided evidence that stock prices are positively related to industrial production but negatively related to the money supply. We also found that for our second vector, normalised on industrial production, that industrial production was negatively related to the interest rate and the rate of inflation. An explanation of the difference in behaviour between the two stock markets may lie in Japan’s slump after 1990 an d its consequent liquidity trap of the late 1990s and early 21st century.外文题目：Can macroeconomic variables explain long term stock market movements?A comparison of the US and Japan出处：School of Economics and Finance, University of St Andrews, St Andrews, UK,作者：Andreas Humpe Peter Macmillan*译文：宏观经济变量可以解释长期的股市走势？一个美国和日本比较研究在一个标准的贴现值模型的框架内，我们考察宏观经济变量是否影响美国和日本的股票价格。

河北省邯郸市武安市武安市第一中学2024-2025学年高二上学期10月期中英语试题一、听力选择题1．What does the woman hurry to do?A．Go to the classroom.B．Catch the bus.C．Go up one floor.2．What does the woman think of the medicine?A．It doesn’t work.B．It makes her tired.C．It makes her have no appetite. 3．How many cookies were left?A．Three.B．Four.C．Ten.4．What may the speakers do next?A．Have a conference.B．Exchange seats.C．Take the fight.5．Where might the speakers be?A．In a library.B．At the man’s home.C．In a supermarket.听下面一段较长对话，回答以下小题。

6．What does the woman like about the old design?A．The red walls.B．The piano.C．The floor.7．What will the speakers do next?A．Play the piano.B．Have something to eat.C．Enjoy the live music.听下面一段较长对话，回答以下小题。

8．What are the speakers doing?A．Having a party B．Travelling to France.C．Watching a movie. 9．Who is the woman speaking to?A．Her brother.B．Her friend.C．Her father.听下面一段较长对话，回答以下小题。

T
• If we know one, then the rule of one market determines that we know the other
6
Rule of One Price: No Arbitrage Profits
Purchase Actual Au Sell T-Bill Sell Au Forward
15
U.K.
•150 ¥ /£
£ 100 (Invested)
9%£ /£
15450 ¥ 15450 ¥ (Repaid)
Forward ¥ /£
£ 109 (Matures)
The Foreign Exchange Parity Relation
• We used the diagram to show that
9
14.8 The “Implied” Risk-Free Rate
• Rearranging the formula, the implied interest rate on a forward given the spot is
F F S0 r S 1; if T 1, r S 0 0
S1 S 0 S1 F h rAu rAu ( syn ) rf F 1 rf h S0 S0 S0
5
Spot-Futures Price Parity for Gold
• A contract with life T: F 1 r f h S 0 • This is not a causal relationship, but the forward and current spot jointly determine the market

1.2 Participants of the Spot Market
• Customers, mainly large companies, require foreign currency in the course of doing business or making investments. • Other types of customers are retail Clients includes smaller companies, individuals who buy foreign exchange to travel abroad or make purchases in foreign countries.
1.1 Introduction:
• Spot transaction: the foreign exchange transactions in the form of bank drafts, checks issued by a bank drawn on different currency-denominated bank accounts.
1.2 The organization of the Spot Market 1.3 The characteristics of the – Small spread: less than 0.1% interbank Market
• The efficiency of the foreign exchange market is revealed in the extremely narrow spreads between buying and selling prices. These spreads can be smaller than a tenth of a percent of the value of a contract .

• We want to ensure that the disturbances are I(0). Under what circumstances will this be the case?
Definition of Cointegration (Engle & Granger, 1987)
Testing for Cointegration in Regression
• The model for the equilibrium correction term can be generalised to include more than two variables: yt = 1 + 2x2t + 3x3t + … + kxkt + ut (3)
Let
zt i X i,t i 1 Then zt I(max di)
k
(1)
Linear Combinations of Non-stationary Variables
• Rearranging (1), we can write
X 1,t i X i ,t z 't
i 2 k
z i i , z 't t , i 2,..., k wherression equation.
• But the disturbances would have some very undesirable properties: zt´ is not stationary and is autocorrelated if all of the Xi are I(1).
• More generally, if we combine variables with differing orders of integration, the combination will have an order of integration equal to the largest. i.e., if Xi,t I(di) for i = 1,2,3,...,k so we have k variables each integrated of order di.

Commitment to sell a commodity at some future date if the purchaser exercises the option
- Max[0, ST – K]
F - ST
Max[0, ST – K] – FV(PC)
-Max[0, ST – K] + FV(PC)
Put-call parity: Call(K,T) – Put(K,T) = PV(F0,T – K)
Purchase Call Option with Strike Price K1 and Sell Call Option with Strike Price K2, where K2>K1
{max[0, ST – K] – FV(PC)} +
{-max[0, K - ST] + FV(PP)}
Mimics long forward position, but involves premiums and uses “strike price” rather than “forward price”
Name Long Forward
Graph
Short Forward
See above
Long Call (Purchased Call)
Short Call (Written Call)
Long Put (Purchased Put)
Dents
Profit graph is identical to that of a purchased call
Payoff graphs can be made identical by adding a zero-coupon bond to the purchased call

CHAPTER 2Futures Markets and Central CounterpartiesPractice QuestionsProblem 2.1.Distinguish between the terms open interest and trading volume.The open interest of a futures contract at a particular time is the total number of long positions outstanding. (Equivalently, it is the total number of short positions outstanding.) The trading volume during a certain period of time is the number of contracts traded during this period. Problem 2.2.What is the difference between a local and a futures commission merchant?A futures commission merchant trades on behalf of a client and charges a commission. A local trades on his or her own behalf.Problem 2.3.Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in the futures price will lead to a margin call? What happens if you do not meet the margin call?There will be a margin call when $1,000 has been lost from the margin account. This will occur when the price of silver increases by 1,000/5,000 = $0.20. The price of silver must therefore rise to $17.40 per ounce for there to be a margin call. If the margin call is not met, your broker closes out your position.Problem 2.4.Suppose that in September 2018 a company takes a long position in a contract on May 2019 crude oil futures. It closes out its position in March 2019. The futures price (per barrel) is $48.30 when it enters into the contract, $50.50 when it closes out its position, and $49.10 at the end of December 2018. One contract is for the delivery of 1,000 barrels. What is the company’s total profit? When is it realized? How is it taxed if it is (a) a hedger and (b) a speculator? Assume that the company has a December 31 year-end.The total profit is ($50.50 - $48.30) ⨯ 1,000 = $2,200. Of this ($49.10 - $48.30) ⨯ 1,000 or $800 is realized on a day-by-day basis between September 2018 and December 31, 2018. A further ($50.50 - $49.10) ⨯ 1,000 or $1,400 is realized on a day-by-day basis between January1, 2019, and March 2019. A hedger would be taxed on the whole profit of $2,200 in 2019. A speculator would be taxed on $800 in 2018 and $1,400 in 2019.Problem 2.5.What does a stop order to sell at $2 mean? When might it be used? What does a limit order to sell at $2 mean? When might it be used?A stop order to sell at $2 is an order to sell at the best available price once a price of $2 or less is reached. It could be used to limit the losses from an existing long position. A limit order to sell at $2 is an order to sell at a price of $2 or more. It could be used to instruct a broker that a short position should be taken, providing it can be done at a price at least as favorable as $2. Problem 2.6.What is the difference between the operation of the margin accounts administered by a clearing house and those administered by a broker?The margin account administered by the clearing house is marked to market daily, and the clearing house member is required to bring the account back up to the prescribed level daily. The margin account administered by the broker is also marked to market daily. However, the account does not have to be brought up to the initial margin level on a daily basis. It has to be brought up to the initial margin level when the balance in the account falls below the maintenance margin level. The maintenance margin is usually about 75% of the initial margin.Problem 2.7.What differences exist in the way prices are quoted in the foreign exchange futures market, the foreign exchange spot market, and the foreign exchange forward market?In futures markets, prices are quoted as the number of U.S. dollars per unit of foreign currency. Spot and forward rates are quoted in this way for the British pound, euro, Australian dollar, and New Zealand dollar. For other major currencies, spot and forward rates are quoted as the number of units of foreign currency per U.S. dollar.Problem 2.8.The party with a short position in a futures contract sometimes has options as to the precise asset that will be delivered, where delivery will take place, when delivery will take place, and so on. Do these options increase or decrease the futures price? Explain your reasoning.These options make the contract less attractive to the party with the long position and more attractive to the party with the short position. They therefore tend to reduce the futures price.Problem 2.9.What are the most important aspects of the design of a new futures contract?The most important aspects of the design of a new futures contract are the specification of the underlying asset, the size of the contract, the delivery arrangements, and the delivery months. Problem 2.10.Explain how margin accounts protect futures traders against the possibility of default.Margin is money deposited by a trader with his or her broker. It acts as a guarantee that the trader can cover any losses on the futures contract. The balance in the margin account is adjusted daily to reflect gains and losses on the futures contract. If losses lead to the balance in the margin account falling below a certain level, the trader is required to deposit a further margin. This system makes it unlikely that the trader will default. A similar system of margin accounts makes it unlikely that the trader’s broker will default on the contract it has with the clearing house member and unlikely that the clearing house member will default with the clearing house. Problem 2.11.A trader buys two July futures contracts on frozen orange juice concentrate. Each contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial margin is $6,000 per contract, and the maintenance margin is $4,500 per contract. What price change would lead to a margin call? Under what circumstances could $2,000 be withdrawn from the margin account?There is a margin call if more than $1,500 is lost on one contract. This happens if the futures price of frozen orange juice falls by more than 10 cents to below 150 cents per pound. $2,000 can be withdrawn from the margin account if there is a gain on one contract of $1,000. This will happen if the futures price rises by 6.67 cents to 166.67 cents per pound.Problem 2.12.Show that, if the futures price of a commodity is greater than the spot price during the delivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if the futures price is less than the spot price? Explain your answer.If the futures price is greater than the spot price during the delivery period, an arbitrageur buys the asset, shorts a futures contract, and makes delivery for an immediate profit. If the futures price is less than the spot price during the delivery period, there is no similar perfect arbitrage strategy. An arbitrageur can take a long futures position but cannot force immediate delivery of the asset. The decision on when delivery will be made is made by the party with the short position. Nevertheless companies interested in acquiring the asset may find it attractive to enterinto a long futures contract and wait for delivery to be made.Problem 2.13.Explain the difference between a market-if-touched order and a stop order.A market-if-touched order is executed at the best available price after a trade occurs at a specified price or at a price more favorable than the specified price. A stop order is executed at the best available price after there is a bid or offer at the specified price or at a price less favorable than the specified price.Problem 2.14.Explain what a stop-limit order to sell at 20.30 with a limit of 20.10 means.A stop-limit order to sell at 20.30 with a limit of 20.10 means that as soon as there is a bid at20.30 the contract should be sold providing this can be done at 20.10 or a higher price. Problem 2.15.At the end of one day a clearing house member is long 100 contracts, and the settlement price is $50,000 per contract. The original margin is $2,000 per contract. On the following day the member becomes responsible for clearing an additional 20 long contracts, entered into at a price of $51,000 per contract. The settlement price at the end of this day is $50,200. How much does the member have to add to its margin account with the exchange clearing house?The clearing house member is required to provide 20$2000$40000⨯,=, as initial margin for the new contracts. There is a gain of (50,200 - 50,000)⨯ 100 = $20,000 on the existing contracts. There is also a loss of (5100050200)20$16000,-,⨯=, on the new contracts. The member must therefore add,-,+,=,400002000016000$36000to the margin account.Problem 2.16.Explain why collateral requirements will increase in the OTC market as a result of new regulations introduced since the 2008 credit crisis.Regulations require most standard OTC transactions entered into between financial institutions to be cleared by CCPs. These have initial and variation margin requirements similar to exchanges. There is also a requirement that initial and variation margin be provided for most bilaterally cleared OTC transactions between financial institutions.As more transactions go through CCPs there will more netting. (Transactions with counterparty A can be netted against transactions with counterparty B providing both are cleared through the same CCP.) This will tend to offset the increase in collateral somewhat. However, it is expected that on balance there will be collateral increases.Problem 2.17.The forward price on the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures price for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes. Which is more favorable for a trader wanting to sell Swiss francs?The 1.1000 forward quote is the number of Swiss francs per dollar. The 0.9000 futures quote is the number of dollars per Swiss franc. When quoted in the same way as the futures price the /.=.. The Swiss franc is therefore more valuable in the forward forward price is11100009091market than in the futures market. The forward market is therefore more attractive for a trader wanting to sell Swiss francs.Problem 2.18.Suppose you call your broker and issue instructions to sell one July hogs contract. Describe what happens.Live hog futures are traded by the CME Group. The broker will request some initial margin. The order will be relayed by telephone to your broker’s trading desk on the floor of the exchange (or to the trading desk of another broker). It will then be sent by messenger to a commission broker who will execute the trade according to your instructions. Confirmation of the trade eventually reaches you. If there are adverse movements in the futures price your broker may contact you to request additional margin.Problem 2.19.“Speculation in futures markets is pure gambling. It is not in the public interest to allow speculators to trade on a futures exchange.” Discuss this viewpoint.Speculators are important market participants because they add liquidity to the market. However, regulators generally only approve contracts when they are likely to be of interest to hedgers as well as speculators.Problem 2.20.Explain the difference between bilateral and central clearing for OTC derivatives.In bilateral clearing the two sides enter into a master agreement covering all outstanding transactions. The agreement defines the circumstances under which transactions can be closedout by one side, how transactions will be valued if there is a close out, how the collateral required by each side is calculated, and so on. In central clearing a CCP stands between the two sides in the same way that an exchange clearing house stands between two sides for transactions entered into on an exchange. The CCP requires initial and variation margin.Problem 2.21.What do you think would happen if an exchange started trading a contract in which the qualityof the underlying asset was incompletely specified?The contract would not be a success. Parties with short positions would hold their contracts until delivery and then deliver the cheapest form of the asset. This might well be viewed by the party with the long position as garbage! Once news of the quality problem became widely known no one would be prepared to buy the contract. This shows that futures contracts are feasible only when there are rigorous standards within an industry for defining the quality of the asset. Many futures contracts have in practice failed because of the problem of defining quality.Problem 2.22.“When a futures contract is traded on the floor of the exchange, it may be the case that the open interest increases by one, stays the same, or decreases by one.” Explain this statement.If both sides of the transaction are entering into a new contract, the open interest increases by one. If both sides of the transaction are closing out existing positions, the open interest decreases by one. If one party is entering into a new contract while the other party is closing out an existing position, the open interest stays the same.Problem 2.23.Suppose that on October 24, 2018, a company sells one April 2019 live-cattle futures contracts. It closes out its position on January 21, 2019. The futures price (per pound) is 121.20 cents when it enters into the contract, 118.30 cents when it closes out its position, and 118.80 cents at the end of December 2018. One contract is for the delivery of 40,000 pounds of cattle. What is the total profit? How is it taxed if the company is (a) a hedger and (b) a speculator?Assume that the company has a December 31 year end.The total profit is40,000×(1.2120−1.1830) = $1,160If the company is a hedger this is all taxed in 2019. If it is a speculator40,000×(1.2120−1.1880) = $960is taxed in 2018 and40,000×(1.1880−1.1830) = $200is taxed in 2019.Problem 2.24.A cattle farmer expects to have 120,000 pounds of live cattle to sell in three months. The live-cattle futures contract traded by the CME Group is for the delivery of 40,000 pounds of cattle. How can the farmer use the contract for hedging? From the farmer’s viewpoint, what are the pros and cons of hedging?The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the gain on the futures contract will offset the loss on the sale of the cattle. If the price of cattle rises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Using futures contracts to hedge has the advantage that the farmer can greatly reduce the uncertainty about the price that will be received. Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices.Problem 2.25.It is July 2017. A mining company has just discovered a small deposit of gold. It will take six months to construct the mine. The gold will then be extracted on a more or less continuous basis for one year. Futures contracts on gold are available with delivery months every two months from August 2017 to December 2018. Each contract is for the delivery of 100 ounces. Discuss how the mining company might use futures markets for hedging.The mining company can estimate its production on a month by month basis. It can then short futures contracts to lock in the price received for the gold. For example, if a total of 3,000 ounces are expected to be produced in September 2018 and October 2018, the price received for this production can be hedged by shorting 30 October 2018 contracts.Problem 2.26.Explain how CCPs work. What are the advantages to the financial system of requiring CCPs to be used for all standard derivatives transactions between financial institutions.?A CCP stands between the two parties in an OTC derivative transaction in much the same way that a clearing house does for exchange-traded contracts. It absorbs the credit risk but requires initial and variation margin from each side. In addition, CCP members are required to contribute to a default fund. The advantage to the financial system is that there is a lot more collateral (i.e., margin) available and it is therefore much less likely that a default by one major participant in the derivatives market will lead to losses by other market participants. The disadvantage is that CCPs are replacing banks as the too-big-to-fail entities in the financial system. There clearly needs to be careful oversight of the management of CCPs.Further QuestionsProblem 2.27.Trader A enters into futures contracts to buy 1 million euros for 1.1 million dollars in three months. Trader B enters in a forward contract to do the same thing. The exchange rate (dollars per euro) declines sharply during the first two months and then increases for the third month to close at 1.1300. Ignoring daily settlement, what is the total profit of each trader? When the impact of daily settlement is taken into account, which trader does better?The total profit of each trader in d ollars is 0.03×1,000,000 = 30,000. Trader B’s profit is realized at the end of the three months. Trader A’s profit is realized day-by-day during the three months. Substantial losses are made during the first two months and profits are made during the final month. It is likely that Trader B has done better because Trader A had to finance its losses during the first two months.Problem 2.28.Explain what is meant by open interest. Why does the open interest usually decline during the month preceding the delivery month? On a particular day, there were 2,000 trades in a particular futures contract. This means that there were 2000 buyers (going long) and 2000 sellers (going short). Of the 2,000 buyers, 1,400 were closing out positions and 600 were entering into new positions. Of the 2,000 sellers, 1,200 were closing out positions and 800 were entering into new positions. What is the impact of the day's trading on open interest?Open interest is the number of contract outstanding. Many traders close out their positions just before the delivery month is reached. This is why the open interest declines during the month preceding the delivery month. The open interest went down by 600. We can see this in two ways. First, 1,400 shorts closed out and there were 800 new shorts. Second, 1,200 longs closed out and there were 600 new longs.Problem 2.29.One orange juice future contract is on 15,000 pounds of frozen concentrate. Suppose that in September 2017 a company sells a March 2019 orange juice futures contract for 120 cents per pound. At the end of December 2017 the futures price is 140 cents; at the end of December 2018 the futures price is 110 cents; and in February 2019 it is closed out at 125 cents. The company has a December 31 year end. What is the company's profit or loss on the contract? How is it realized? What is the accounting and tax treatment of the transaction if the company is classified as a) a hedger and b) a speculator?The price goes up during the time the company holds the contract from 120 to 125 cents per pound. Overall the company therefore takes a loss of 15,000×0.05 = $750. If the company is classified as a hedger this loss is realized in 2019, If it is classified as a speculator it realizes aloss of 15,000×0.20 = $3000 in 2017, a gain of 15,000×0.30 = $4,500 in 2018, and a loss of 15,000×0.15 = $2,250 in 2019.Problem 2.30.A company enters into a short futures contract to sell 5,000 bushels of wheat for 750 cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin call? Under what circumstances could $1,500 be withdrawn from the margin account?There is a margin call if $1000 is lost on the contract. This will happen if the price of wheat futures rises by 20 cents from 750 cents to 770 cents per bushel. $1500 can be withdrawn if the futures price falls by 30 cents to 720 cents per bushel.Problem 2.31.Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is 4% per annum. How could you make money if the June and December futures contracts for a particular year trade at $50 and $56?You could go long one June oil contract and short one December contract. In June you take delivery of the oil borrowing $50 per barrel at 4% to meet cash outflows. The interest accumulated in six months is about 50×0.04×1/2 or $1 per barrel. In December the oil is sold for $56 per barrel which is more than the $51 that has to be repaid on the loan. The strategy therefore leads to a profit. Note that this profit is independent of the actual price of oil in June and December. It will be slightly affected by the daily settlement procedures.Problem 2.32.What position is equivalent to a long forward contract to buy an asset at K on a certain date and a put option to sell it for K on that date?The long forward contract provides a payoff of S T−K where S T is the asset price on the date and K is the delivery price. The put option provides a payoff of max (K−S T, 0). If S T> K the sum of the two payoffs is S T–K. If S T< K the sum of the two payoffs is 0. The combined payoff is therefore max (S T–K, 0). This is the payoff from a call option. The equivalent position is therefore a call option.Problem 2.33.A company has derivatives transactions with Banks A, B, and C which are worth +$20 million, −$15 million, and −$25 million, respectively to t he company. How much margin or collateral does the company have to provide in each of the following two situations?a) The transactions are cleared bilaterally and are subject to one-way collateral agreements where the company posts variation margin, but no initial margin. The banks do not have to post collateral.b) The transactions are cleared centrally through the same CCP and the CCP requires a total initial margin of $10 million.If the transactions are cleared bilaterally, the company has to provide collateral to Banks A, B, and C of (in millions of dollars) 0, 15, and 25, respectively. The total collateral required is $40 million. If the transactions are cleared centrally they are netted against each other and the company’s total variation margin (i n millions of dollars) is –20 + 15 + 25 or $20 million in total. The total margin required (including the initial margin) is therefore $30 million.Problem 2.34.A bank’s derivatives transactions with a counterparty are worth +$10 million to the bankand are cleared bilaterally. The counterparty has posted $10 million of cash collateral.What credit exposure does the bank have?The counterparty may stop posting collateral and some time will then elapse before the bank is able to close out the transaction s. During that time the transactions may move in the bank’s favor, increasing its exposure. Note that the bank is likely to have hedged the transactions and will incur a loss on the hedge if the transactions move in the bank’s favor. For example, if the transactions change in value from $10 to $13 million after the counterparty stops posting collateral, the bank loses $3 million on the hedge and will not necessarily realize an offsetting gain on the transactions.Problem 2.35. (Excel file)The author’s We b page (www-2.rotman.utoronto.ca/~hull/data) contains daily closing prices for crude oil futures contract and gold futures contract. You are required to download the data for crude oil and answer the following:(a) Assuming that daily price changes are normally distributed with zero mean, estimate the standard deviation of daily price changes. Calculate the standard deviation of two-day changes from the standard deviation of one-day changes assuming that changes are independent.(b) Suppose that an exchange wants to set the margin requirement for a member with a long position in one contract so that it is 99% certain that the margin will not be wiped out by a two-day price move. (It chooses two days because it considers that it can take two days to close out a defaulting member.) How high does the margin have to be when the normal distribution assumption is made? Each contract is on 1,000 barrels of oil.(c) Use the data to determine how often the margin of the member would actually be wiped out by a two-day price move. What do your results suggest about the appropriateness of the normal distribution assumption?(d) Suppose that for retail clients the maintenance margin is equal to the amount calculated in (b) and is 75% of the initial margin. How frequently would the balance in the account of a client with a long position be negative immediately before a margin payment is due (so that the client has an incentive to default)? Assume that balances in excess of the initial margin are withdrawn by the client.(a)contract. The standard deviation of two-day price changes is $1577.7contract.(b)Margin for member =$2,231.2×2.326 = $5,190.6(c)Worksheet shows that the margin would be wiped out 24 times or on 2.31% of the days.This suggests that price changes have heavier tails than the normal distribution.(d)Worksheet shows that there would be 157 margin calls and the client has an incentive todefault 9 times.。

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Yap and Reinsel (1995), Saikkonen and Luukkonen (1997), Quintos (1998), Stock and Watson (1988), Hansen and Phillips (1990), Engle and Yoo (1991), Phillips (1991), Phillips and Loretan (1991), Saikkonen (1991), Bewley and Yang (1995), Yang and Bewley (1995) and Bierens (1997) are some of these proposals. The main characteristic that share all these procedures is that they specify the null hypothesis of no cointegration against the alternative hypothesis of cointegration. However, this specication has been criticised in Engle and Granger (1987), Phillips and Ouliaris (1990) and Engle and Yoo (1991) where it is argued that the natural specication to be tested should be the null hypothesis of cointegration instead of the null hypothesis of lack of cointegration. Papers like the ones of Hansen (1992b), Leybourne and McCabe (1993), Harris and Inder (1994), Shin (1994) and McCabe, Leybourne and Shin (1997) have addressed this question reversing these hypotheses and designing tests for the null hypothesis of cointegration. In fact, the denition of cointegration cannot be disassociated from the long-run. The cointegration property implies that a set of variables moves jointly dening an equilibrium relationship which means that it has to be tested using time series covering long periods. Practitioners have to be aware to have in mind that the larger sample period covering the time series the more probability of appearance of a structural change. As shown by Perron (1989), this possibility changes the distribution of the standard unit root tests since the specication for the deterministic component of the time series needs to be modied to collect the effect of the structural breaks. In these terms, the denition of cointegration under consideration falls into the classication of deterministic and stochastic cointegration of Park (1990). The generalization of Perron’s proposal to the cointegration tests of Engle and Granger (1987) and Phillips and Ouliaris (1990) is done by Gregory and Hansen (1996a, 1996b). On the other hand, Hao (1996) generalises the test of Kwiatkowski, Phillips, Schmidt and Shin (1992) to allow for a structural break that shifts the independent term of the cointegrating vector. In a recent paper, Bartley, Lee and Strazicich (2001) deal also with the KPSS test but now including a shift in the mean and in the trend of the deterministic elements of the cointegrating relationship.1 In the present paper we extend the test of the null hypothesis of cointegration to allow for a structural break in both the parameters of the deterministic component and the parameters of the stochastic component. Our proposal generalises the previous contributions and develops a test for cointegration around a break-cointegrating relationship. We remark the differences that can be established between these previous contributions and the ones proposed here. We also consider the endogenous selection of the time break, analysing up to three different methods to estimate it. In section 2 we present the models and test statistics and derive their asymptotic distribution under two assumptions: (i) the stochastic regressors are strictly exogenous and (ii) the break point is assumed to be known. In section 3 we relax the rst assumption and present the way to get efcient estimates of the cointegrating vector. Section 4 deals with the estimation of the break point. Section 5 shows the consistency of the test statistics proposed in section 2. Section 6 analyses the performance of the tests in nite

Forward exchange rateNot to be confused with forward rate or forward price. The forward exchange rate(also referred to as for-ward rate or forward price)is the exchange rate at which a bank agrees to exchange one currency for an-other at a future date when it enters into a forward contract with an investor.[1][2][3]Multinational corpora-tions,banks,and otherfinancial institutions enter into forward contracts to take advantage of the forward rate for hedging purposes.[1]The forward exchange rate is determined by a parity relationship among the spot ex-change rate and differences in interest rates between two countries,which reflects an economic equilibrium in the foreign exchange market under which arbitrage opportu-nities are eliminated.When in equilibrium,and when interest rates vary across two countries,the parity con-dition implies that the forward rate includes a premium or discount reflecting the interest rate differential.For-ward exchange rates have important theoretical implica-tions for forecasting future spot exchange rates.Financial economists have put forth a hypothesis that the forward rate accurately predicts the future spot rate,for which empirical evidence is mixed.1IntroductionThe forward exchange rate is the rate at which a com-mercial bank is willing to commit to exchange one cur-rency for another at some specified future date.[1]The forward exchange rate is a type of forward price.It is the exchange rate negotiated today between a bank and a client upon entering into a forward contract agreeing to buy or sell some amount of foreign currency in the future.[2][3]Multinational corporations andfinancial in-stitutions often use the forward market to hedge future payables or receivables denominated in a foreign cur-rency against foreign exchange risk by using a forward contract to lock in a forward exchange rate.Hedging with forward contracts is typically used for larger transac-tions,while futures contracts are used for smaller transac-tions.This is due to the customization afforded to banks by forward contracts traded over-the-counter,versus the standardization of futures contracts which are traded on an exchange.[1]Banks typically quote forward rates for major currencies in maturities of one,three,six,nine, or twelve months,however in some cases quotations for greater maturities are available up tofive or ten years.[2]2Relation to covered interest rate parityCovered interest rate parity is a no-arbitrage condition in foreign exchange markets which depends on the availabil-ity of the forward market.It can be rearranged to give the forward exchange rate as a function of the other variables. The forward exchange rate depends on three known vari-ables:the spot exchange rate,the domestic interest rate, and the foreign interest rate.This effectively means that the forward rate is the price of a forward contract,which derives its value from the pricing of spot contracts and the addition of information on available interest rates.[4] The following equation represents covered interest rate parity,a condition under which investors eliminate expo-sure to foreign exchange risk(unanticipated changes in exchange rates)with the use of a forward contract–the exchange rate risk is effectively covered.Under this con-dition,a domestic investor would earn equal returns from investing in domestic assets or converting currency at the spot exchange rate,investing in foreign currency assets in a country with a different interest rate,and exchanging the foreign currency for domestic currency at the nego-tiated forward exchange rate.Investors will be indiffer-ent to the interest rates on deposits in these countries due to the equilibrium resulting from the forward exchange rate.The condition allows for no arbitrage opportunities because the return on domestic deposits,1+id,is equal to the return on foreign deposits,[S/F](1+if).If these two returns weren't equalized by the use of a forward con-tract,there would be a potential arbitrage opportunity in which,for example,an investor could borrow currency in the country with the lower interest rate,convert to the foreign currency at today’s spot exchange rate,and invest in the foreign country with the higher interest rate.[4] (1+i d)=SF(1+i f)whereF is the forward exchange rateS is the current spot exchange rateid is the interest rate in domestic currency(basecurrency)if is the interest rate in foreign currency(quoted currency)124FORECASTING FUTURE SPOT EXCHANGE RATES This equation can be arranged such that it solves for theforward rate:F=S (1+i f) (1+i d)3Forward premium or discount The equilibrium that results from the relationship be-tween forward and spot exchange rates within the con-text of covered interest rate parity is responsible for elim-inating or correcting for market inefficiencies that would create potential for arbitrage profits.As such,arbitrage opportunities arefleeting.In order for this equilibrium to hold under differences in interest rates between two countries,the forward exchange rate must generally dif-fer from the spot exchange rate,such that a no-arbitrage condition is sustained.Therefore,the forward rate is said to contain a premium or discount,reflecting the interest rate differential between two countries.The following equations demonstrate how the forward premium or dis-count is calculated.[1][2]The forward exchange rate differs by a premium or dis-count of the spot exchange rate:F=S(1+P)whereP is the premium(if positive)or discount(ifnegative)The equation can be rearranged as follows to solve for the forward premium/discount:P=FS−1In practice,forward premiums and discounts are quoted as annualized percentage deviations from the spot ex-change rate,in which case it is necessary to account for the number of days to delivery as in the following example.[2]P N=(FS−1)360dwhereN represents the maturity of a given forward exchange rate quoted represents the number of days to delivery For example,to calculate the6-month forward premium or discount for the euro versus the dollar deliverable in30 days,given a spot rate quote of1.2238$/€and a6-month forward rate quote of1.2260$/€:P6=(1.22601.2238−1)36030=0.021572=2.16%The resulting0.021572is positive,so one would say that the euro is trading at a0.021572or2.16%premium against the dollar for delivery in30days.Conversely,if one were to work this example in euro terms rather than dollar terms,the perspective would be reversed and one would say that the dollar is trading at a discount against the Euro.4Forecasting future spot exchange rates4.1Unbiasedness hypothesisThe unbiasedness hypothesis states that given conditions of rational expectations and risk neutrality,the forward exchange rate is an unbiased predictor of the future spot exchange rate.Without introducing a foreign exchange risk premium(due to the assumption of risk neutral-ity),the following equation illustrates the unbiasedness hypothesis.[3][5][6][7]F t=E t(S t+k)whereF t is the forward exchange rate at time tE t(S t+k)is the expected future spot exchangerate at time t+kk is the number of periods into the future fromtime tThe empirical rejection of the unbiasedness hypothesis is a well-recognized puzzle amongfinance researchers. Empirical evidence for cointegration between the forward rate and the future spot rate is mixed.[5][8][9]Researchers have published papers demonstrating empirical failure of the hypothesis by conducting regression analyses of the realized changes in spot exchange rates on forward premi-ums andfinding negative slope coefficients.[10]These re-searchers offer numerous rationales for such failure.One rationale centers around the relaxation of risk neutrality, while still assuming rational expectations,such that a for-eign exchange risk premium may exist that can account for differences between the forward rate and the future spot rate.[11]3The following equation represents the forward rate as be-ing equal to a future spot rate and a risk premium(not tobe confused with a forward premium):[12]F t=E t(S t+1)+P tThe current spot rate can be introduced so that the equa-tion solves for the forward-spot differential(the differ-ence between the forward rate and the current spot rate): F t−S t=E t(S t+1−S t)+P tEugene Fama concluded that large positive correlationsof the difference between the forward exchange rate andthe current spot exchange rate signal variations over timein the premium component of the forward-spot differen-tial F t−S t or in the forecast of the expected change in the spot exchange rate.Fama suggested that slope coef-ficients in the regressions of the difference between theforward rate and the future spot rate F t−S t+1,and the expected change in the spot rate E t(S t+1−S t),on the forward-spot differential F t−S t which are different from zero imply variations over time in both components of the forward-spot differential:the premium and the expected change in the spot rate.[12]Fama’sfindings were sought to be empirically validated by a significant body of re-search,ultimatelyfinding that large variance in expected changes in the spot rate could only be accounted for by risk aversion coefficients that were deemed“unaccept-ably high.”[7][11]Other researchers have found that the unbiasedness hypothesis has been rejected in both cases where there is evidence of risk premia varying over time and cases where risk premia are constant.[13]Other rationales for the failure of the forward rate unbi-asedness hypothesis include considering the conditionalbias to be an exogenous variable explained by a pol-icy aimed at smoothing interest rates and stabilizing ex-change rates,or considering that an economy allowing fordiscrete changes could facilitate excess returns in the for-ward market.Some researchers have contested empiri-cal failures of the hypothesis and have sought to explainconflicting evidence as resulting from contaminated dataand even inappropriate selections of the time length offorward contracts.[11]Economists demonstrated that theforward rate could serve as a useful proxy for future spotexchange rates between currencies with liquidity premiathat average out to zero during the onset offloating ex-change rate regimes in the1970s.[14]Research examiningthe introduction of endogenous breaks to test the struc-tural stability of cointegrated spot and forward exchangerate time series have found some evidence to support for-ward rate unbiasedness in both the short and long term.[9] 5See also•Foreign exchange derivative 6References[1]Madura,Jeff(2007).International Financial Manage-ment:Abridged8th Edition.Mason,OH:Thomson South-Western.ISBN0-324-36563-2.[2]Eun,Cheol S.;Resnick,Bruce G.(2011).InternationalFinancial Management,6th Edition.New York,NY: McGraw-Hill/Irwin.ISBN978-0-07-803465-7.[3]Levi,Maurice D.(2005).International Finance,4th Edi-tion.New York,NY:Routledge.ISBN978-0-415-30900-4.[4]Feenstra,Robert C.;Taylor,Alan M.(2008).Interna-tional Macroeconomics.New York,NY:Worth Publish-ers.ISBN978-1-4292-0691-4.[5]Delcoure,Natalya;Barkoulas,John;Baum,ChristopherF.;Chakraborty,Atreya(2003).“The Forward RateUnbiasedness Hypothesis Reexamined:Evidence froma New Test”.Global Finance Journal14(1):83–93.doi:10.1016/S1044-0283(03)00006-1.Retrieved2011-06-21.[6]Ho,Tsung-Wu(2003).“A re-examination of the un-biasedness forward rate hypothesis using dynamic SUR model”.The Quarterly Review of Economics and Finance 43(3):542–559.doi:10.1016/S1062-9769(02)00171-0.Retrieved2011-06-23.[7]Sosvilla-Rivero,Simón;Park,Young B.(1992).“Furthertests on the forward exchange rate unbiasedness hy-pothesis”.Economics Letters40(3):325–331.doi:10.1016/0165-1765(92)90013-O.Retrieved2011-06-27.[8]Moffett,Michael H.;Stonehill,Arthur I.;Eiteman,DavidK.(2009).Fundamentals of Multinational Finance,3rd Edition.Boston,MA:Addison-Wesley.ISBN978-0-321-54164-2.[9]Villanueva,O.Miguel(2007).“Spot-forward cointegra-tion,structural breaks and FX market unbiasedness”.In-ternational Financial Markets,Institutions&Money17(1):58–78.doi:10.1016/j.intfin.2005.08.007.Retrieved2011-06-22.[10]Zivot,Eric(2000).“Cointegration and forward andspot exchange rate regressions”.Journal of In-ternational Money and Finance19(6):785–812.doi:10.1016/S0261-5606(00)00031-0.Retrieved2011-06-22.[11]Diamandis,Panayiotis F.;Georgoutsos,Dimitris A.;Kouretas,Georgios P.(2008).“Testing the forward rate unbiasedness hypothesis during the1920s”.International Financial Markets,Institutions&Money18(4):358–373.doi:10.1016/j.intfin.2007.04.003.Retrieved2011-06-23.[12]Fama,Eugene F.(1984).“Forward and spot exchangerates”.Journal of Monetary Economics14(3):319–338.doi:10.1016/0304-3932(84)90046-1.Retrieved2011-06-20.[13]Chatterjee,Devalina(2010).Three essays in forward rateunbiasedness hypothesis(Thesis).Utah State University.pp.1–102.Retrieved2012-06-21.46REFERENCES [14]Cornell,Bradford(1977).“Spot rates,forward rates andexchange market efficiency”.Journal of Financial Eco-nomics5(1):55–65.doi:10.1016/0304-405X(77)90029-0.Retrieved2011-06-28.5 7Text and image sources,contributors,and licenses7.1Text•Forward exchange rate 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Journal of Economic Perspectives—Volume15,Number4—Fall2001—Pages101–115 Vector AutoregressionsJames H.Stock and Mark W.WatsonM acroeconometricians do four things:describe and summarize macro-economic data,make macroeconomic forecasts,quantify what we do ordo not know about the true structure of the macroeconomy,and advise (and sometimes become)macroeconomic policymakers.In the1970s,these four tasks—data description,forecasting,structural inference and policy analysis—were performed using a variety of techniques.These ranged from large models with hundreds of equations to single-equation models that focused on interactions of a few variables to simple univariate time series models involving only a single variable. But after the macroeconomic chaos of the1970s,none of these approaches appeared especially trustworthy.Two decades ago,Christopher Sims(1980)provided a new macroeconometric framework that held great promise:vector autoregressions(VARs).A univariate autoregression is a single-equation,single-variable linear model in which the cur-rent value of a variable is explained by its own lagged values.A VAR is an n-equation,n-variable linear model in which each variable is in turn explained by its own lagged values,plus current and past values of the remaining nϪ1variables. This simple framework provides a systematic way to capture rich dynamics in multiple time series,and the statistical toolkit that came with VARs was easy to use and to interpret.As Sims(1980)and others argued in a series of influential early papers,VARs held out the promise of providing a coherent and credible approach to data description,forecasting,structural inference and policy analysis.In this article,we assess how well VARs have addressed these four macroecono-y James H.Stock is the Roy rsen Professor of Political Economy,John F.Kennedy School of Government,Harvard University,Cambridge,Massachusetts.Mark W.Watson is Profes-sor of Economics and Public Affairs,Department of Economics and Woodrow Wilson School of Public and International Affairs,Princeton University,Princeton,New Jersey.Both authors are Research Associates,National Bureau of Economic Research,Cambridge,Massachusetts.102Journal of Economic Perspectivesmetric tasks.1Our answer is“it depends.”In data description and forecasting,VARs have proven to be powerful and reliable tools that are now,rightly,in everyday use. Structural inference and policy analysis are,however,inherently more difficult because they require differentiating between correlation and causation;this is the “identification problem,”in the jargon of econometrics.This problem cannot be solved by a purely statistical tool,even a powerful one like a VAR.Rather,economic theory or institutional knowledge is required to solve the identification(causation versus correlation)problem.A Peek Inside the VAR ToolkitWhat,precisely,is the effect of a100-basis-point hike in the federal funds interest rate on the rate of inflation one year hence?How big an interest rate cut is needed to offset an expected half percentage point rise in the unemployment rate?How well does the Phillips curve predict inflation?What fraction of the variation in inflation in the past40years is due to monetary policy as opposed to external shocks?Many macroeconomists like to think they know the answers to these and similar questions,perhaps with a modest range of uncertainty.In the next two sections,we take a quantitative look at these and related questions using several three-variable VARs estimated using quarterly U.S.data on the rate of price inflation(␲t),the unemployment rate(u t)and the interest rate(R t,specifically,the federal funds rate)from1960:I–2000:IV.2First,we construct and examine these models as a way to display the VAR toolkit;criticisms are reserved for the next section.VARs come in three varieties:reduced form,recursive and structural.A reduced form VAR expresses each variable as a linear function of its own past values,the past values of all other variables being considered and a serially uncor-related error term.Thus,in our example,the VAR involves three equations: current unemployment as a function of past values of unemployment,inflation and the interest rate;inflation as a function of past values of inflation,unemployment and the interest rate;and similarly for the interest rate equation.Each equation is estimated by ordinary least squares regression.The number of lagged values to include in each equation can be determined by a number of different methods, and we will use four lags in our examples.3The error terms in these regressions are the“surprise”movements in the variables after taking its past values into account. If the different variables are correlated with each other—as they typically are in 1Readers interested in more detail than provided in this brief tutorial should see Hamilton’s(1994) textbook or Watson’s(1994)survey article.ϭ400ln(P t/P tϪ1),where P t is the chain-weighted GDP price 2The inflation data are computed as␲tindex and u t is the civilian unemployment rate.Quarterly data on u t and R t are formed by taking quarterly averages of their monthly values.3Frequently,the Akaike(AIC)or Bayes(BIC)information criteria are used;for a discussion,see Lu¨tkepohl(1993,chapter4).James H.Stock and Mark W.Watson103 macroeconomic applications—then the error terms in the reduced form model will also be correlated across equations.A recursive VAR constructs the error terms in each regression equation to be uncorrelated with the error in the preceding equations.This is done by judiciously including some contemporaneous values as regressors.Consider a three-variable VAR,ordered as1)inflation,2)the unemployment rate,and3)the interest rate. In thefirst equation of the corresponding recursive VAR,inflation is the dependent variable,and the regressors are lagged values of all three variables.In the second equation,the unemployment rate is the dependent variable,and the regressors are lags of all three variables plus the current value of the inflation rate.The interest rate is the dependent variable in the third equation,and the regressors are lags of all three variables,the current value of the inflation rate plus the current value of the unemployment rate.Estimation of each equation by ordinary least squares produces residuals that are uncorrelated across equations.4Evidently,the results depend on the order of the variables:changing the order changes the VAR equations,coefficients,and residuals,and there are n!recursive VARs representing all possible orderings.A structural VAR uses economic theory to sort out the contemporaneous links among the variables(Bernanke,1986;Blanchard and Watson,1986;Sims,1986). Structural VARs require“identifying assumptions”that allow correlations to be interpreted causally.These identifying assumptions can involve the entire VAR,so that all of the causal links in the model are spelled out,or just a single equation,so that only a specific causal link is identified.This produces instrumental variables that permit the contemporaneous links to be estimated using instrumental vari-ables regression.The number of structural VARs is limited only by the inventiveness of the researcher.In our three-variable example,we consider two related structural VARs.Each incorporates a different assumption that identifies the causal influence of monetary policy on unemployment,inflation and interest rates.Thefirst relies on a version of the“Taylor rule,”in which the Federal Reserve is modeled as setting the interest rate based on past rates of inflation and unemployment.5In this system,the Fed sets the federal funds rate R according to the ruleR tϭr*ϩ1.5͑␲៮tϪ␲*͒Ϫ1.25͑u៮tϪu*͒ϩlagged values of R,␲,uϩ␧t, where r*is the desired real rate of interest,␲៮t and u៮t are the average values of inflation and unemployment rate over the past four quarters,␲*and u*are the target values of inflation and unemployment,and␧t is the error in the equation. This relationship becomes the interest rate equation in the structural VAR.4In the jargon of VARs,this algorithm for estimating the recursive VAR coefficients is equivalent to estimating the reduced form,then computing the Cholesky factorization of the reduced form VAR covariance matrix;see Lu¨tkepohl(1993,chapter2).5Taylor’s(1993)original rule used the output gap instead of the unemployment rate.Our version uses Okun’s Law(with a coefficient of2.5)to replace the output gap with unemployment rate.104Journal of Economic PerspectivesThe equation error,␧t,can be thought of as a monetary policy“shock,”sinceit represents the extent to which actual interest rates deviate from this Taylor rule.This shock can be estimated by a regression with R tϪ1.5␲៮tϩ1.25u៮t as the dependent variable,and a constant and lags of interest rates,unemployment andinflation on the right-hand side.The Taylor rule is“backward looking”in the sense that the Fed reacts to pastinformation(␲៮t and u៮t are averages of the past four quarters of inflation and unemployment),and several researchers have argued that Fed behavior is more appropriately described by forward-looking behavior.Because of this,we consider another variant of the model in which the Fed reacts to forecasts of inflation and unemployment four quarters in the future.This Taylor rule has the same form as the rule above,but with␲៮t and u៮t replaced by four-quarter ahead forecasts com-puted from the reduced form VAR.Putting the Three-Variable VAR Through Its PacesThe different versions of the inflation-unemployment-interest rate VAR are put through their paces by applying them to the four macroeconometric tasks. First,the reduced form VAR and a recursive VAR are used to summarize the comovements of these three series.Second,the reduced form VAR is used to forecast the variables,and its performance is assessed against some alternative benchmark models.Third,the two different structural VARs are used to estimate the effect of a policy-induced surprise move in the federal funds interest rate on future rates of inflation and unemployment.Finally,we discuss how the structural VAR could be used for policy analysis.Data DescriptionStandard practice in VAR analysis is to report results from Granger-causality tests,impulse responses and forecast error variance decompositions.These statistics are computed automatically(or nearly so)by many econometrics packages(RATS, Eviews,TSP and others).Because of the complicated dynamics in the VAR,these statistics are more informative than are the estimated VAR regression coefficients or R2statistics,which typically go unreported.Granger-causality statistics examine whether lagged values of one variable help to predict another variable.For example,if the unemployment rate does not help predict inflation,then the coefficients on the lags of unemployment will all be zero in the reduced-form inflation equation.Panel A of Table1summarizes the Granger-causality results for the three-variable VAR.It shows the p-values associated with the F-statistics for testing whether the relevant sets of coefficients are zero.The unemployment rate helps to predict inflation at the5percent significance level (the p-value is0.02,or2percent),but the federal funds interest rate does not(the p-value is0.27).Inflation does not help to predict the unemployment rate,but the federal funds rate does.Both inflation and the unemployment rates help predict the federal funds interest rate.Table1VAR Descriptive Statistics for(␲,u,R)A.Granger-Causality TestsRegressorDependent Variable in Regression␲u R␲0.000.310.00 u0.020.000.00 R0.270.010.00 B.Variance Decompositions from the Recursive VAR Ordered as␲,u,RB.i.Variance Decomposition of␲Forecast HorizonForecastStandard ErrorVariance Decomposition(Percentage Points)␲u R10.9610000 4 1.3488102 8 1.758217112 1.9782162B.ii.Variance Decomposition of uForecast HorizonForecastStandard ErrorVariance Decomposition(Percentage Points)␲u R10.231990 40.640982 80.7978211 120.92166618B.iii.Variance Decomposition of RForecast HorizonForecastStandard ErrorVariance Decomposition(Percentage Points)␲u R10.85219794 1.84950418 2.4412602812 2.63165925Notes:␲denotes the rate of price inflation,u denotes the unemploy-ment rate and R denotes the Federal Funds interest rate.The entriesin Panel A show the p-values for F-tests that lags of the variable in therow labeled Regressor do not enter the reduced form equation for thecolumn variable labeled Dependent Variable.The results were com-puted from a VAR with four lags and a constant term over the1960:I–2000:IV sample period.Vector Autoregressions105106Journal of Economic PerspectivesImpulse responses trace out the response of current and future values of each of the variables to a one-unit increase in the current value of one of the VAR errors, assuming that this error returns to zero in subsequent periods and that all other errors are equal to zero.The implied thought experiment of changing one error while holding the others constant makes most sense when the errors are uncorre-lated across equations,so impulse responses are typically calculated for recursive and structural VARs.The impulse responses for the recursive VAR,ordered␲t,u t,R t,are plotted in Figure1.Thefirst row shows the effect of an unexpected1percentage point increase in inflation on all three variables,as it works through the recursive VAR system with the coefficients estimated from actual data.The second row shows the effect of an unexpected increase of1percentage point in the unemployment rate, and the third row shows the corresponding effect for the interest rate.Also plotted areϮ1standard error bands,which yield an approximate66percent confidence interval for each of the impulse responses.These estimated impulse responses show patterns of persistent common variation.For example,an unexpected rise in inflation slowly fades away over24quarters and is associated with a persistent increase in unemployment and interest rates.The forecast error decomposition is the percentage of the variance of the error made in forecasting a variable(say,inflation)due to a specific shock(say,the error term in the unemployment equation)at a given horizon(like two years).Thus,the forecast error decomposition is like a partial R2for the forecast error,by forecast horizon.These are shown in Panel B of Table1for the recursive VAR.They suggest considerable interaction among the variables.For example,at the12-quarter horizon,75percent of the error in the forecast of the federal funds interest rate is attributed to the inflation and unemployment shocks in the recursive VAR. ForecastingMultistep-ahead forecasts,computed by iterating forward the reduced form VAR,are assessed in Table2.Because the ultimate test of a forecasting model is its out-of-sample performance,Table2focuses on pseudo out-of-sample forecasts over the period from1985:I to2000:IV.It examines forecast horizons of two quarters, four quarters and eight quarters.The forecast h steps ahead is computed by estimating the VAR through a given quarter,making the forecast h steps ahead, reestimating the VAR through the next quarter,making the next forecast and so on through the forecast period.6As a comparison,pseudo out-of-sample forecasts were also computed for a univariate autoregression with four lags—that is,a regression of the variable on lags 6Forecasts like these are often referred to as pseudo or“simulated”out-of-sample forecasts to emphasize that they simulate how these forecasts would have been computed in real time,although,of course,this exercise is conducted retrospectively,not in real time.Our experiment deviates slightly from what would have been computed in real time because we use the current data,which includes later revisions made to the inflation and unemployment data by statistical agencies,rather than the data available in real time.of its own past values—and for a random walk (or “no change”)forecast.Inflation rate forecasts were made for the average value of inflation over the forecast period,while forecasts for the unemployment rate and interest rate were made for the final quarter of the forecast period.Table 2shows the root mean square forecast error for each of the forecasting methods.(The mean squared forecast error is computed as the average squared value of the forecast error over the 1985–2000out-of-sample period,and the resulting square root is the root mean squared forecast error reported in the table.)Table 2indicates that the VAR either does no worse than or improves upon the univariate autoregression and that both improve upon the random walk forecast.Structural InferenceWhat is the effect on the rates of inflation and unemployment of a surprise 100basis–point increase in the federal funds interest rate?Translated into VAR jargon,Figure 1Impulse Responses in the Inflation-Unemployment-Interest Rate RecursiveVAR James H.Stock and Mark W.Watson 107this question becomes:What are the impulse responses of the rates of inflation and unemployment to the monetary policy shock in a structural VAR?The solid line in Figure 2plots the impulse responses computed from our model with the backward-looking Taylor rule.It shows the inflation,unemploy-ment and real interest rate (R t Ϫ␲t )responses to a 1percentage point shock in the nominal federal funds rate.The initial rate hike results in the real interest rate exceeding 50basis points for six quarters.Although inflation is eventually reduced by approximately 0.3percentage points,the lags are long,and most of the action occurs in the third year after the contraction.Similarly,the rate of unemployment rises by approximately 0.2percentage points,but most of the economic slowdown is in the third year after the rate hike.How sensitive are these results to the specific identifying assumption used in this structural VAR—that the Fed follows the backward-looking Taylor rule?As it happens,very sensitive.The dashed line in Figure 2plots the impulse responses computed from the structural VAR with the forward-looking Taylor rule.The impulse responses in real interest rates are broadly similar under either rule.However,in the forward-looking model the monetary shock produces a 0.5per-centage point increase in the unemployment rate within a year,and the rate of inflation drops sharply at first,fluctuates,then leaves a net decline of 0.5percent-age points after six years.Under the backward-looking rule,this 100basis-point rate hike produces a mild economic slowdown and a modest decline in inflation several years hence;under the forward-looking rule,by this same action the Fed wins a major victory against inflation at the cost of a swift and sharp recession.Policy AnalysisIn principle,our small structural VAR can be used to analyze two types of policies:surprise monetary policy interventions and changing the policy rule,like shifting from a Taylor rule (with weight on both unemployment and inflation)to an explicit inflation targeting rule.Table 2Root Mean Squared Errors of Simulated Out-Of-Sample Forecasts,1985:1–2000:IVForecastHorizonInflation RateUnemployment Rate Interest Rate RW AR VAR RW AR VAR RW AR VAR 2quarters0.820.700.680.340.280.290.790.770.684quarters0.730.650.630.620.520.53 1.36 1.25 1.078quarters 0.750.750.75 1.120.950.78 2.18 1.92 1.70Notes:Entries are the root mean squared error of forecasts computed recursively for univariate and vector autoregressions (each with four lags)and a random walk (“no change”)model.Results for the random walk and univariate autoregressions are shown in columns labeled RW and AR,respectively.Each model was estimated using data from 1960:I through the beginning of the forecast period.Forecasts for the inflation rate are for the average value of inflation over the period.Forecasts for the unemployment rate and interest rate are for the final quarter of the forecast period.108Journal of Economic PerspectivesIf the intervention is an unexpected movement in the federal funds interest rate,then the estimated effect of this policy on future rates of inflation and unemployment is summarized by the impulse response functions plotted in Figure2.This might seem a somewhat odd policy,but the same mechanics can be used to evaluate a more realistic intervention,such as raising the federal funds rate by 50basis points and sustaining this increase for one year.This policy can be engineered in a VAR by using the right sequence of monetary policy innovations to hold the federal funds interest rate at this sustained level for four quarters,taking into account that in the VAR,actions on interest rates in earlier quarters affect those in later quarters (Sims,1982;Waggoner and Zha,1999).Analysis of the second type of policy—a shift in the monetary rule itself—is more complicated.One way to evaluate a new policy rule candidate is to ask what would be the effect of monetary and nonmonetary shocks on the economy under the new rule.Since this question involves all the structural disturbances,answering Figure 2Impulse Responses of Monetary Policy Shocks for Different Taylor Rule IdentifyingAssumptionsNotes:The solid line is computed with the backward-looking Taylor rule;the dashed line,with the forward-looking Taylor rule.Vector Autoregressions 109110Journal of Economic Perspectivesit requires a complete macroeconomic model of the simultaneous determination of all the variables,and this means that all of the causal links in the structural VAR must be specified.In this case,policy analysis is carried out as follows:a structural VAR is estimated in which all the equations are identified,then a new model is formed by replacing the monetary policy paring the impulse responses in the two models shows how the change in policy has altered the effects of monetary and nonmonetary shocks on the variables in the model.How Well Do VARs Perform the Four Tasks?We now turn to an assessment of VARs in performing the four macroecono-metric tasks,highlighting both successes and shortcomings.Data DescriptionBecause VARs involve current and lagged values of multiple time series,they capture comovements that cannot be detected in univariate or bivariate models. Standard VAR summary statistics like Granger-causality tests,impulse response functions and variance decompositions are well-accepted and widely used methods for portraying these comovements.These summary statistics are useful because they provide targets for theoretical macroeconomic models.For example,a theoretical model that implied that interest rates should Granger-cause inflation but unem-ployment should not would be inconsistent with the evidence in Table1.Of course,the VAR methods outlined here have some limitations.One is that the standard methods of statistical inference(such as computing standard errors for impulse responses)may give misleading results if some of the variables are highly persistent.7Another limitation is that,without modification,standard VARs miss nonlinearities,conditional heteroskedasticity and drifts or breaks in parameters.ForecastingSmall VARs like our three-variable system have become a benchmark against which new forecasting systems are judged.But while useful as a benchmark,small VARs of two or three variables are often unstable and thus poor predictors of the future(Stock and Watson,1996).State-of-the-art VAR forecasting systems contain more than three variables and allow for time-varying parameters to capture important drifts in coefficients(Sims, 1993).However,adding variables to the VAR creates complications,because the number of VAR parameters increases as the square of the number of variables:a nine-variable,four-lag VAR has333unknown coefficients(including the inter-7Bootstrap methods provide some improvements(Kilian,1999)for inference about impulse responses, but treatments of this problem that are fully satisfactory theoretically are elusive(Stock,1997;Wright, 2000).James H.Stock and Mark W.Watson111 cepts).Unfortunately,macroeconomic time series data cannot provide reliable estimates of all these coefficients without further restrictions.One way to control the number of parameters in large VAR models is to impose a common structure on the coefficients,for example using Bayesian meth-ods,an approach pioneered by Litterman(1986)(six variables)and Sims(1993) (nine variables).These efforts have paid off,and these forecasting systems have solid real-time track records(McNees,1990;Zarnowitz and Braun,1993). Structural InferenceIn our three-variable VAR in the previous section,the estimated effects of a monetary policy shock on the rates of inflation and unemployment(summarized by the impulse responses in Figure2)depend on the details of the presumed mone-tary policy rule followed by the Federal Reserve.Even modest changes in the assumed rule resulted in substantial changes in these impulse responses.In other words,the estimates of the structural impulse responses hinge on detailed institu-tional knowledge of how the Fed sets interest rates.8Of course,the observation that results depend on assumptions is hardly new. The operative question is whether the assumptions made in VAR models are any more compelling than in other econometric models.This is a matter of heated debate and is thoughtfully discussed by Leeper,Sims and Zha(1996),Christiano, Eichenbaum and Evans(1999),Cochrane(1998),Rudebusch(1998)and Sims (1998).Below are three important criticisms of structural VAR modeling.9 First,what really makes up the VAR“shocks?”In large part,these shocks,like those in conventional regression,reflect factors omitted from the model.If these factors are correlated with the included variables,then the VAR estimates will contain omitted variable bias.For example,officials at the Federal Reserve might scoff at the idea that they mechanically followed a Taylor rule,or any other fixed-coefficient mechanical rule involving only a few variables;rather,they suggest that their decisions are based on a subtle analysis of very many macroeconomic factors,both quantitative and qualitative.These considerations,when omitted from the VAR,end up in the error term and(incorrectly)become part of the estimated historical“shock”used to estimate an impulse response.A concrete example of this in the VAR literature involves the“price puzzle.”Early VARs showed an odd result: inflation tended to increase following monetary policy tightening.One explanation for this(Sims,1992)was that the Fed was looking forward when it set interest rates and that simple VARs omitted variables that could be used to predict future inflation.When these omitted variables intimated an increase in inflation,the Fed tended to increase interest rates.Thus,these VAR interest rate shocks presaged 8In addition,the institutional knowledge embodied in our three-variable VAR is rather naı¨ve;for example,the Taylor rule was designed to summarize policy in the Greenspan era,not the full sample in our paper.9This list hits only the highlights;other issues include the problem of“weak instruments”discussed in Pagan and Robertson(1998)and the problem of noninvertible representations discussed in Hansen and Sargent(1991)and Lippi and Reichlin(1993).112Journal of Economic Perspectivesincreases in inflation.Because of omitted variables,the VAR mistakenly labeled these increases in interest rates as monetary shocks,which led to biased impulse responses.Indeed,Sims’s explanation of the price puzzle has led to the practice of including commodity prices in VARs to attempt to control for predicted future inflation.Second,policy rules change over time,and formal statistical tests reveal widespread instability in low-dimensional VARs(Stock and Watson,1996).Con-stant parameter structural VARs that miss this instability are improperly identified. For example,several researchers have documented instability in monetary policy rules(for example,Bernanke and Blinder,1992;Bernanke and Mihov,1998; Clarida,Gali and Gertler,2000;Boivin,2000),and this suggests misspecification in constant coefficient VAR models(like our three-variable example)that are esti-mated over long sample periods.Third,the timing conventions in VARs do not necessarily reflect real-time data availability,and this undercuts the common method of identifying restrictions based on timing assumptions.For example,a common assumption made in struc-tural VARs is that variables like output and inflation are sticky and do not respond “within the period”to monetary policy shocks.This seems plausible over the period of a single day,but becomes less plausible over a month or quarter.In this discussion,we have carefully distinguished between recursive and structural VARs:recursive VARs use an arbitrary mechanical method to model contemporaneous correlation in the variables,while structural VARs use economic theory to associate these correlations with causal relationships.Unfortunately,in the empirical literature the distinction is often murky.It is tempting to develop economic“theories”that,conveniently,lead to a particular recursive ordering of the variables,so that their“structural”VAR simplifies to a recursive VAR,a structure called a“Wold causal chain.”We think researchers yield to this temptation far too often.Such cobbled-together theories,even if superficially plausible,often fall apart on deeper inspection.Rarely does it add value to repackage a recursive VAR and sell it as structural.Despite these criticisms,we think it is possible to have credible identifying assumptions in a VAR.One approach is to exploit detailed institutional knowledge. An example of this is the study by Blanchard and Perotti(1999)of the macroeco-nomic effects offiscal policy.They argue that the tax code and spending rules impose tight constraints on the way that taxes and spending vary within the quarter, and they use these constraints to identify the exogenous changes in taxes and spending necessary for causal analysis.Another example is Bernanke and Mihov (1998),who use a model of the reserves market to identify monetary policy shocks.A different approach to identification is to use long-run restrictions to identify shocks;for example,King,Plosser,Stock and Watson(1991)use the long-run neutrality of money to identify monetary shocks.However,assumptions based on the infinite future raise questions of their own(Faust and Leeper,1997).A constructive approach is to recognize explicitly the uncertainty in the assumptions that underlie structural VAR analysis and see what inferences,or range of inferences,still can be made.For example,Faust(1998)and Uhlig(1999)。

Cointegrationand...*Thanks to Charles Engel for his insights on the issues presented herein and to VinayDatar for motivating me to write the paper. Updates to the paper can be found at /doc/62d52332b90d6c85ec3ac6cd.html /~ezivot/homepage.htm.Cointegration and Forward and Spot Exchange Rate RegressionsbyEric Zivot *Department of Economics University of WashingtonBox 353330Seattle, WA 98195-3330ezivot@/doc/62d52332b90d6c85ec3 ac6cd.html June 12, 1997Last Revision: September 29, 1998AbstractIn this paper we investigate in detail the relationship between models of cointegration between the current spot exchange rate, s t , and the current forward rate, f t , and models of cointegration between the future spot rate, s t+1, and f t and the implications of this relationship for tests of the forward rate unbiasedness hypothesis (FRUH). We argue that simple models of cointegration between s t and f t more easily capture the stylized facts of typical exchange rate data than simple models of cointegration between s t+1 and f t and so serve as a natural starting point for the analysis of exchange rate behavior. Weshow that simple models of cointegration between s t and f t imply rather complicated models of cointegration between s t+1 and f t . As a result, standard methods are often not appropriate for modeling the cointegrated behavior of (s t+1, f t )N and we show that the use of such methods can lead to erroneous inferences regarding the FRUH.21. IntroductionThere is an enormous literature on testing if the forward exchange rate is an unbiased predictor of future spot exchange rates. Engel (1996) provides the most recent review. The earliest studies, e.g. Cornell (1977), Levich (1979) and Frenkel (1980), were based on the regression of the log of the future spot rate, s t+1, on the log of the current forward rate, f t . The results of these studies generally support the forward rate unbiasedness hypothesis (FRUH). Due to the unit root behavior of exchange rates and the concern about the spurious regression phenomenon illustrated by Granger and Newbold (1974), later studies, e.g. Bilson (1981), Fama (1984) and Froot and Frankel (1989),concentrated on the regression of the change in the log spot rate, )s t+1, on the forward premium, f t -s t . Overwhelmingly, the results of these studies reject the FRUH. The most recent studies, e.g.,Hakkio and Rush (1989), Barnhart and Szakmary (1991), Naka and Whitney (1995), Hai, Mark and Yu (1997), Norrbin and Reffett (1996), Newbold, et. al. (1996), Clarida and Taylor (1997), Barnhart,McNown and Wallace (1998) and Luintel and Paudyal (1998) have focused on the relationship between cointegration and tests of the FRUH. The results of these studies are mixed and depend on how cointegration is modeled.Since the results of Hakkio and Rush (1989), it is wellrecognized that the FRUH requires that s t+1 and f t be cointegrated and that the cointegrating vector be (1,-1) and much of the recent literature has utilized models of cointegration between s t+1 and f t . It is also true that the FRUH requires s t and f t to be cointegrated with cointegrating vector (1,-1) and only a few authors have based their analysis on models of cointegration between s t and f t . In this paper we investigate in detail the relationship between models of cointegration between s t and f t and models of cointegration between s t+1 and f t and the implications of this relationship for tests of the FRUH. We argue that simple models of cointegration between s t and f t more easily capture the stylized facts of typical exchange rate data than simple models of cointegration between s t+1 and f t and so serve as a natural starting point for the analysis of exchange rate behavior. Simple models of cointegration between s t and f t imply rather complicated models of cointegration between s t+1 and f t . In particular, starting with a first order bivariate vector error correction model for (s t , f t )N we show that the implied cointegrated model for (s t+1, f t )N is nonstandard and does not have a finite VAR representation. As a result,standard VAR methods are not appropriate for modeling (s t+1, f t )N and we show that the use of such3methods can lead to erroneous inferences regarding the FRUH. In particular, we show that tests of the null of no-cointegration based on common cointegrated models for (s t+1, f t )N are likely to be severely size distorted. In addition, using the implied triangular cointegrated representation for (s t+1,f t )N we can explicitly characterize the OLS bias in the levels regression of s t+1 on f t . Based on this representation, we can show that theOLS estimate of the coefficient on f t in the levels regression is downward biased (away from one) even if the FRUH is true. Finally, we show that the results of Naka and Whitney (1995) and Norrbin and Reffett (1996) supporting the FRUH are driven by the specification of the cointegration model for (s t+1, f t )N .The plan of the paper is as follows. In section 2, we discuss the relationship between cointegration and the tests of the forward rate unbiasedness hypothesis. In section 3 we present some stylized facts of exchange rate data typically used in investigations of the FRUH. In section 4, we discuss some simple models of cointegration between s t and f t that capture the basic stylized facts about the data and we show the restrictions that the FRUH places on these models. In section 5, we consider models of cointegration between s t+1 and f t that are implied by models of cointegration between s t and f t . In section 6, we use our results to reinterpret some recent findings concerning the FRUH reported by Naka and Whitney (1995) and Norrbin and Reffett (1996). Our concluding remarks are given in section 7.2. Cointegration and the Forward Rate Unbiasedness Hypothesis: An OverviewThe relationship between cointegration and the forward rate unbiasedness hypothesis has been discussed by several authors starting with Hakkio and Rush (1987). Engel (1996) provides a comprehensive review of this literature and serves as a starting point for the analysis in this paper.Following Engel (1996), the forward exchange rate unbiasedness hypothesis (FRUH) under rational expectations and risk neutrality is given byE t [s t+1] = f t ,(1)where E t [@] denotes expectation conditional oninformation available at time t. Using the terminology of Baillie (1989), FRUH is an example of the “observable expectations” hypothesis. The FRUH is usually expressed as the levels relationship4s t+1 = f t + >t+1,(2)where >t+1 is a random variable (rational expectations forecast error) with E t [>t+1] = 0. It should be kept in mind that rejection of the FRUH can be interpreted as a rejection of the model underlying E t [@]or a rejection of the equality in (1) itself.Two different regression equations have generally been used to test the FRUH. The first is the “levels regression”s t+1 = μ + $f f t + u t+1(3)and the null hypothesis that FRUH is true imposes the restrictions μ = 0, $f = 1 and E t [u t+1] = 0. In early empirical applications authors generally focused on testing the first two conditions and either ignored the latter condition or informally tested it using the Durbin-Watson statistic. Most studies using (3) found estimates of $f very close to 1 and hence supported the FRUH. Some authors, e.g.Barnhart and Szakmary (1991), Liu and Maddala (1992), Naka and Whitney (1995) and Hai, Mark and Wu (1997), refer to testing μ = 0, $f = 1 as tes ting the forward rate unbiasedness condition (FRUC). Testing the orthogonality condition E t [u t+1] = 0, conditional on not rejecting FRUC, is then referred to as testing forward market efficiency under rational expectaions and risk neutrality.Assuming s t and f t have unit roots, i.e., s t , f t ~ I (1), (see, for example, Messe and Singleton (1982),Baillie and Bollerslev (1989), Mark (1990), Liuand Maddala (1992), Crowder (1994), or Clarida and Taylor (1997) for empirical evidence), then the FRUH requires that s t+1 and f t be cointegrated with cointegrating vector (1, -1) and that the stationary, i.e., I (0), cointegrating residual, u t+1, satisfy E t [u t+1] = 0. Notice that testing FRUC is then equivalent to testing for cointegration between s t+1 and f t and that the cointegrating vector is (1,-1) and testing forward market efficiency is equivalent to testing that the forecast error, s t+1 - f t , has conditional mean zero.The FRUH assumes rational expectations and risk neutrality. Under rational expectations,if agents are risk averse then a stationary time-varying risk premium exists and the relationship between s t+1 and f t becomess t+1 = f t - rp t re + >t+1,(4)where rp t re = f t - E t [s t+1] represents the stationary rational expectations risk premium. As long as rp treis stationary s t+1 and f t will be cointegrated with cointegrating vector (1, -1) but f t will be a biased predictor of s t+1 provided rp t re is predictable using information available at time t . In this regard, the5FRUC is a misleading acronym since if the FRUC is true forward rates are not necessarily unbiased predictors of future spot rates. Since the FRUC is equivalent to cointegration between s t+1 and f t and that the cointegrating vector is (1,-1), which implies that s t+1 and f t trend together in the long-run but may deviate in the short-run, it is more appropriate to call this thelong-run forward rate unbiasedness condition (LRFRUC).Several authors, e.g. Messe and Singleton (1982), Meese (1989) and Isard (1995), have stated that since s t and f t have unit roots the levels regression (3) is not a valid regression equation because of the spurious regression problem described in Granger and Newbold (1974). However,this is not true if s t+1 and f t are cointegrated. What is true is that if s t+1 and f t are cointegrated with cointegrating vector (1, -1), which allows for the possibility of a stationary time varying risk premium so that u t+1 in (3) is I (0), then the OLS estimates from (3) will be super consistent (converge at rate T instead of rate T ?) for the true value $f = 1 but generally not efficient and biased away from 1 in finite samples so that the asymptotic distributions of t -tests and F -tests on μ an $f will follow non-standard distributions, see Corbae, Lim and Ouliaris (1992). Hence, even if the FRUH is not true due the existence of a stationary risk premium so that (3) is a misspecified regression, OLS on (3) still gives a consistent estimate of $f = 1. More importantly, there are simple modifications to OLS that yield asymptotically unbiased and efficient estimates of the parameters of (3) in the presence of general serial correlation and heteroskedasticity and these modifications should be used to make inferences about the parameters in the levels regression 1. In this sense, the levels regression (3) under cointegration is asymptotically immune to the omission of a stationary risk premium. Hai, Mark and Wu (1997) use Stock and Watson’s (1993) dynamic OLS estimator on the levels regression (3) and provide evidence that s t+1 and f t are cointegrated with cointegrating vector (1, -1).The second regression equation used to test the FRUH is the “differences equation”)s t+1 = μ* + "s (f t - s t ) + u*t+1(5)and the null hypothesis that FRUH is true imposes the restrictions μ* = 0, "s = 1 and E t [u*t+1] = 0.Empirical results based on (5), surveyed in Engel (1996), overwhelmingly reject the FRUH. In fact,typical estimates of "s across a wide range of currencies and sampling frequencies are significantly negative. This result is often referred to as the forward discount anomaly, forward discount bias or forward discount puzzle and seems to contradict the results based on the levels regression (3). Given 6that s t , f t ~ I (1), for (5) to be a “balanced regression” (i.e., all variables in the regression are integrated of the same order) the forward premium, f t - s t , must be I (0) or, equivalently, f t and s t must be cointegrated with cointegrating vector (1,-1). Assuming that covered interest rate parity holds, the forward premium is simply the interest rate differential between the respective countries and there are good economic reasons to believe that such differentials do not contain a unit root. Hence, tests of the FRUH based on (5) implicitly assume that the forward premium is I (0) and so such tests are conditional on f t and s t being cointegrated with cointegrating vector (1,-1). In this respect, (5) can be thought of as one equation in a particular vector error correction model (VECM) for (f t , s t )N .2Horvath and Watson (1995) and Clarida and Taylor (1997) use VECM-based tests and provide evidence that f t and s t are cointegrated with cointegrating vector (1,-1).As noted by Fama (1984), the negative estimates of "s are consistent with rationalexpectations and market efficiency and imply certainrestrictions on the risk premium. T o see this,note that under rational expectations we may write)s t+1=(f t - s t ) - rp t re +>t+1,(6)so that the difference regression (5) is misspecified if risk neutrality fails. Since all variables in (6) are I (0), if the risk premium is correlated with the forward premium then the OLS estimate of "s in the standard differences regression (5), which omits the risk premium, will be biased away from the true value of 1. Hence the negative estimates of "s from (5) can be interpreted as resulting from omitted variables bias. As discussed in Fama (1984), for omitted variables bias to account for negative estimates of "s it must be true that cov(E t [s t+1] - s t , rp t re ) < 0 and var(rp t re) > Var(E t [s t+1] - s t ). Hence,as Engel (1996) notes, models of the foreign exchange risk premium should be consistent with these two inequalities.The tests of the FRUH based on (3) and (5) involve cointegration either between s t+1 and f tor f t and s t . As Engel (1996) points out, sinces t+1 - f t = )s t+1 - (f t - s t )it is trivial to see under the assumption that f t and s t are I(1) that (i) if s t and f t are cointegrated with cointegrating vector (1,-1) then s t+1 and f t must be cointegrated with cointegrating vector (1,-1); and (ii) if s t+1 and f t are cointegrated with cointegrating vector (1,-1) then s t and f t must be cointegrated with cointegrating vector (1,-1). Cointegration models for (s t , f t )N and (s t+1, f t )N can both be used to7describe the data and test the FRUH but the form of themodels used can have a profound impact on the resulting inferences. For example, we show that a simple first order vector error correction model for (s t , f t )N describes monthly data well and leads naturally to the differences regression (5) from which the FRUH is easily rejected. In contrast, we show that some simple first order vector error correction models for (s t+1, f t )N , which are used in the empirical studies of Naka and Whitney (1995) and Norrbin and Reffett (1996), miss some important dynamics in monthly data and as a result indicate that the FRUH appears to hold. Hence, misspecification of the cointegration model for (s t+1, f t )N can explain some of the puzzling empirical results concerning tests of the FRUH.In the next section, we describe some stylized facts of monthly exchange rate data that are typical in the analysis of the FRUH. In the remaining sections we use these facts to motivate certain models of cointegration for (s t , f t )N and (s t+1, f t )N to support our claims regarding misspecification and tests of the FRUH.3. Some Stylized Facts of Typical Exchange Rate DataLet f t denote the log of the forward exchange rate in month t and s t denote the log of the spot exchange rate. We focus on monthly data for which the maturity date of the forward contract is the same as the sampling interval to avoid modeling complications created by overlapping data. For our empirical examples, we consider forward and spot rate data (all relative to the U.S. dollar) on the pound, yen and Canadian dollar taken from Datastream 3. Figure 1 shows time plots of s t+1, f t - s t (forward premium), and s t+1 - f t (forecast error) for the three currencies and Table 1 gives some summary statistics of the data. Spot and forward rates behave very similarly and exhibit random walk typebehavior. The forward premiums are all highly autocorrelated but the forecast errors show very little autocorrelation. The variances of )s t+1 and )f t+1 are roughly ten times larger than the variance of f t - s t and are similar to the variance of s t+1 - f t . For all currencies, )s t+1, )f t+1 and s t+1 - f t are negatively correlated with f t - s t . Any model of cointegration with cointegrating vector (1, -1) for (s t ,f t )N or (s t+1, f t )N should capture these basic stylized facts.4. Models of Cointegration between f t and s t4.1 Vector error correction representation8The stylized facts of the monthly exchange rate data reported in the previous section can be captured by a simple cointegrated VAR(1) model for y t = (f t , s t )N . This simple model has also recently been used by Godbout and van Norden (1996). Before presenting the empirical restults, we begin this section with a review of the properties of such a model. The general bivariate VAR(1) model for y t isy t = μ + M y t-1 + ,t ,where ,t ~ iid (0,E ) and E has elements F ij (i,j = f,s ), and can be reparameterized as)y t = μ + A y t-1+ ,t(7)where A = M - I . Under the assumption of cointegration, A has rank 1 and there exist 2 × 1 vectors " and $ such that A = "$N . Using the normalization $ = (1, -$s )N , (7) becomes the vector error correction model (VECM))f t = μf + "f (f t-1 - $s s t-1) + ,ft ,(8a))s t = μs + "s (f t-1 - $s s t-1) + ,st .(8b)Since spot and forward rates often do not exhibit a systematic tendency to drift up or down it may be more appropriate to restrict the intercepts in (8) to the error correction term. That is, μf = -"f μc and μs = -"s μc . Under this restriction s t and f t are I (1) without drift and the cointegrating residual,f t - $s s t , is allowed to have a nonzero mean μc 4.With the intercepts in (8) restricted to the error correction term, the VECM can be solved to give a simple AR(1) model for the cointegrating residual $N y t - μc = f t - $s s t - μc . Premultiplying (7)by $N and rearranging givesf t - $s s t - μc = N (f t-1 - $s s t-1 - μc ) + 0t ,(9)where N = 1 + $N " = 1 + ("f - $s "s ) and 0t = $N ,t = ,ft - $s ,st . Since (9) is simply an AR(1) model,the cointegrating residual is stable and stationary if *N * = *1 + ("f - $s "s )* < 1. Notice that if "f =$s "s then the cointegrating residual is I (1) and f t and s t are not cointegrated.The exogeneity status of spot and forward rates with regard to the cointegrating parameters " and $ was the focus of attention in Norrbin and Reffett (1996) so it is appropriate to discuss this issue in some detail. Exogeneity issues in error correction models are discussed at length in Johansen (1992, 1995), Banerjee et. al. (1993), Urbain (1993), Ericsson and Irons (1994) and Zivot (1998).For our purposes weak exogeneity of spot or forward rates for the cointegrating parameters " and 9$ places restrictions on the parameters of the VECM (8). In particular, if f t is weakly exogenous with respect to ("s , $s )N then "f = 0 and efficient estimation of the cointegrating parameters can be made from the single equation conditionalerror correction model)s t = μs + "s (f t-1 - $s s t-1) + (s )f t + <="" bdsfid="179" p="">(10a)where (s = F s -s 1 F fs and <st is="" uncorrelated="" with="" ,ft="" .="" similarly,="" if="" s="" t="" weakly="" exogenous="" respect="" to="" ("f="" ,="" $s="" )n="" then="" "s="0" and="" efficient="" estimation="" of="" the="" cointegrating="" parameters="" can="" be="" made="" from="" single="" equation="" conditional="" error="" correction="" model<="" p="" bdsfid="182">。