Cointegration and Forward and Spot Exchange Rate Regressions

*

Thanks to Charles Engel for his insights on the issues presented herein and to Vinay

Datar for motivating me to write the paper. Updates to the paper can be found at https://www.doczj.com/doc/633332328.html,/~ezivot/homepage.htm.

Cointegration and Forward and Spot Exchange Rate Regressions

by

Eric Zivot *

Department of Economics University of Washington

Box 353330

Seattle, WA 98195-3330ezivot@https://www.doczj.com/doc/633332328.html, June 12, 1997

Last Revision: September 29, 1998

Abstract

In 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. We show 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.

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1. Introduction

There 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 well recognized 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 such

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methods 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 the OLS 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 Overview

The 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 by

E t [s t+1] = f t ,

(1)

where E t [@] denotes expectation conditional on information 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 relationship

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s 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 testing 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), Liu and 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 becomes

s 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 t

re

is 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, the

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FRUC 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 the long-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

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that 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 rational

expectations and market efficiency and imply certain restrictions on the risk premium. To 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 t

or f t and s t . As Engel (1996) points out, since

s 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 to

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describe the data and test the FRUH but the form of the models 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 Data

Let 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 type behavior. 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 t

4.1 Vector error correction representation

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The 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 is

y 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 gives

f 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

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$ 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 conditional error correction model

)s t = μs + "s (f t-1 - $s s t-1) + (s )f t +

(10a)

where (s = F s -s 1 F fs and

)f t = μf + "f (f t-1 - $s s t-1) + (f )s t +

(10b)

where (f = F f -f 1 F fs and

If $s = 1 then the forward premium is I (0) and follows an AR(1) process and the VECM (8)

becomes

)f t = μf + "f (f t-1 - s t-1) + ,ft ,(11a))s t = μs + "s (f t-1 - s t-1) + ,st .

(11b)Notice that (11b) is simply the standard differences regression (5) used to test the FRUH. Further,if "f and "s are of the same sign and magnitude then the implied value of N in (9) is close to 1 and this corresponds to the stylized fact that the forward premium is stationary but very highly autocorrelated.Also, the implied variance of the of the forward premium from (9) is F 00 = F ff + F ss - 2D fs (F ff F ss )? and will be very small relative to the variances of )f t and )s t given the stylized facts that F ff . F ss and D fs . 1.

The FRUH places testable restrictions on the VECM (8). Necessary conditions for the FRUH to hold are (i) s t and f t are cointegrated (ii) $s = 1 and (iii) μc = 0. In addition, the FRUH requires that "s = 1 in order for the forecast error in (2) to have conditional mean zero. It is important to stress that, together, these two restrictions limit both the long-run and short-run behavior of spot and forward rates. Applying these restrictions, (8), led one period, becomes

)f t+1 = "f (f t - s t ) + ,f,t+1,(12a))s t+1 = (f t - s t ) + ,s,t+1.

(12b)

Notice that the FRUH requires that the expected change spot rate is equal to the forward premium or, equivalently, that the adjustment to long-run equilibrium occurs in one period. The change in the forward rate, on the other hand, is directly related to the persistence of interest rate differentials now

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measured by "f since N = 1 + ("f - 1) = "f . Stability of the VECM under the FRUH requires that *"f *< 1. Thus, the FRUH is consistent with a highly persistent forward premium.

The representation in (12) shows that weak exogeneity of spot rates with respect to the cointegrating parameters is inconsistent with the FRUH because if spot rates are weakly exogenous then "s = 0 and the FRUH cannot hold. In addition if the FRUH is true and forward rates are weakly exogenous then (12) cannot capture the dynamics of typical data. To see this, suppose that forward rates are weakly exogenous so that "f = 0. Since F ss . F ff . F sf = F 2 it follows that (s . (f . 1. If μs = 0, "s = 1 and $s = 1, then (10a) becomes

s t = f t-1 + )f t +

which simply states that the current spot rate is equal to the current forward rate plus a white noise error. This result is clearly inconsistent with the data since it implies that the forward premium is serially uncorrelated.

4.2 Phillips’ triangular representation .

Another useful representation of a cointegrated system is Phillips’ (1991) triangular representation, which is similar to the triangular representation of a limited information simultaneous equations model. This representation is most useful for studying the asymptotic properties of cointegrating regressions and Baillie (1989) has advocated its use for testing rational expectations restrictions in cointegrated VAR models. This representation is also used by Naka and Whitney (1995) to test the FRUH. The general form of the triangular representation for y t is

f t = μc + $s s t + u ft ,(13a)s t = s t-1 + u st ,

(13b)

where the vector of errors u t = (u ft , u st )N = (f t - $s s t - μc , )s t )N has the stationary moving average

representation u t = R (L)e t where and e t is i.i.d. with mean zero and

R (L )'j 4

k '0R k L k

,j 4

k '0

|k |R k <4covariance matrix V . Equation (13a) models the (structural) cointegrating relationship and (13b) is a reduced form relationship describing the stochastic trend in the spot rate. For a given VECM representation, the triangular representation is simply a reparameterization. For the VECM (8) with the restricted constant, the derived triangular representation is given by (13a)-(13b) with

u ft = N u f,t-1 + 0t ,

(13c)

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u st = "s u ft-1 + ,st ,

(13d)

and N , "s , 0t and ,st are as previously defined. Equation (13c) models the disequilibrium error (which equals the forward premium if $s = 1) as an AR(1) process and (13d) allows the lagged error to affect the change in the spot rate. Let e t = (0t , ,st )N . Then the vector u t = (u ft , u st )N = (f t - $s s t - μc ,)s t )N has the VAR(1) representation u t = Cu t-1 + e t where

C

'

N "s ,V

'

F 00F 0F s 0F ,

Hence, R (L) = (I - C L)-1. As noted previously, F 00 is very small relative to F ss and F 0s = F fs - F ss .

Phillips and Loretan (1991) and Phillips (1991) show how the triangular representation of a cointegrated system can be used to derive the asymptotic properties of the OLS estimates of the cointegration parameters. For our purposes, the most important result is that the OLS estimate of $s from (13a) is asymptotically unbiased and efficient only if u ft and u st are contemporaneously uncorrelated and there is no feedback between u ft and u st (i.e, s t is weakly exogenous for $s and u ft does not Granger cause u st and vise-versa). These correlation and feedback effects can be expressed in terms of specific components of the long-run covariance matrix of u t . The long-run covariance

matrix of u t is defined as = (I - C )-1V (I - C )-1N and this matrix can be

S 'j 4

k '&4E [u 0u )

k ]'R (1)V R (1))decomposed into S = ) + 'N where ) = '0 + ', '0 = E[u 0u 0N ] and . Let S and

''j 4k '1

E [u 0u )

k ]) have elements T ij and )ij (i,j = 0,s ), respectively. Using these matrices it can be shown that the elements that contribute to the bias and nonnormality of OLS estimates are the quantities 2 = T s 0/T ss and )s 0, which measure the long-run correlation and endogeneity between u ft and u st . If these elements are zero then the OLS estimates are asymptotically (mixed) normal, unbiased and efficient https://www.doczj.com/doc/633332328.html,ing the triangular system (13) some tedious calculations show (see Zivot (1995))

2 = ("s F 00/(1 - N ) + F 0,/(1 - N ))@("s 2F 00/(1 - N ) + 2"s F 0s /(1 - N ) + F ss )-1,)s 0 = "s F 00N [(1 - N )(1 - N 2)]-1 + F 0s /(1 - N ),

and these quantities are zero if "s = 0 (spot rates are weakly exogenous for $s ) and F 0s = 0. In typical exchange rate data, however, F 0s . 0 and F 00 is much smaller than F ss which implies that 2 .0 and )s 0 . 0 so the OLS bias is expected to be very small.

To illustrate the expected magnitude of the OLS bias we conducted a simple Monte Carlo

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experiment where data was generated by (13) with $s = 1, "s = 1, -3, N = 0.9, F ss = (0.035)2, F 00 =(0.001)2, and D 0s = 06. When "s = 1 the FRUH is true and when "s = -3 it is not. In both cases the forward premium is highly autocorrelated. Table 2 gives the results of OLS applied to the levels regression (12a) for samples of size T =100 and T =250. In both cases the magnitude of the OLS bias is negligible. The OLS standard errors, however, are biased which cause the size distortions in the nominal 5% t -tests of the hypothesis that $s = 1.4.3 Empirical Example

Table 3 presents estimation results for the VAR model (7) and Table 4 gives the results for the triangular model (13) imposing $s = 1 for the pound, yen and Canadian dollar monthly exchange rate series. The VAR(1) model was selected for all currencies by likelihood ratio tests for lag lengths and standard model selection criteria. For all currencies, f t and s t behave very similarly: the estimated intercepts and error variances are nearly identical and the estimated correlation between ,ft and ,st ,D fs , is 0.99. The estimated coefficients from the triangular model essentially mimic the corresponding coefficients from the VAR(1). Table 5 gives the results of Johansen’s likelihood ratio test for the number of cointegrating vectors for the three exchange rate series based on the estimation of (7). If the intercepts are restricted to the error correction term then the Johansen rank test finds one cointegrating vector in all cases. If the intercept is unrestricted then the rank tests finds that spot and forward rates for the pound and Canadian dollar are I (0). However, for each series, the likelihood ratio statistic does not reject the hypothesis that the intercepts be restricted to the error correction term. Table 5 also reports the results of the Engle-Granger (1987) two-step residual based ADF t -test for no cointegration based on estimating $s by OLS. For long lag lengths the null of no-cointegration between forward and spot rates is not rejected at the 10% level but for short lag lengths the null is rejected at the 5% level 7. Table 6 reports estimates of $s using OLS, Stock and Watson’s (1993)dynamic OLS (DOLS) and dynamic GLS (DGLS) lead-lag estimator and Johansen’s (1995) reduced rank MLE 8. The latter three estimators are asymptotically efficient estimators and yield asymptotically valid standard errors. Notice that all of the estimates of $s are extremely close to 1and the hypothesis that $s = 1 cannot be rejected using the asymptotic t-tests based on DOLS/DGLS and MLE. Table 7 shows the MLEs of the parameters of the VECM (8) where the intercepts are

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restricted to the error correction term. Notice also that the estimates of "f and "s are both significantly negative and of about the same magnitude indicting that the error correction term, which is essentially

the forward premium since $^

s . 1, is very highly autocorrelated. Further, since the estimates of "f

and "s are significantly different from zero neither spot nor forward rates appear to be weakly exogenous with respect to the cointegrating parameters.

The above empirical results are based on a two-step procedure of first testing for cointegration between f t and s t and then testing if the cointegrating vector is (1,-1). Alternatively, one can use a one-step procedure to test the null of no-cointegration against the joint hypothesis of cointegration with a prespecified cointegrating vector. The advantage of using tests that impose a prespecified cointegrating vector is that if the cointegrating vector is true then the test can have substantially higher power than tests that implicitly estimate the cointegrating vector 9. The most commonly used one-step procedure to test the joint hypothesis of cointegration between f t and s t with the prespecified cointegrating vector (1,-1) is to run either a unit root test (e.g. ADF t-test) or a stationarity test (e.g KPSS test) on the forward premium f t - s t . Table 5 also reports the ADF unit root tests and KPSS stationarity tests on the forward premia for the three exchange rate series. For short lag lengths the ADF tests indicate that the forward premia are I (0) whereas for long lags the series appear I (1). The KPSS tests give mixed results 10.

Use of the ADF test as a test for no-cointegration against the alternative of cointegration with a prespecified cointegrating vector, however, has been criticized by Kremers, Ericsson and Dolado (1992) and Zivot (1998) as having low power since the ADF test places unrealistic parametric restrictions on the short-run dynamics of the data. They show that a single equation conditional ECM test, based on models similar to (10a) and (10b), can have substantially higher power than the ADF test. A limitation of the conditional ECM test, however, is that it assumes weak exogeneity of one variable with respect to the cointegration parameters. In the context of testing the stationarity of the forward premium it is hard to argue a priori that either forward rates or spot rates are weakly exogenous, and the empirical results of table 5 indicate they are not, and so it appears that the conditional ECM test is not appropriate. Horvath and Watson (1995), however, develop a multivariate procedure to test for cointegration with a prespecified cointegrating vector within a VECM that does not require any exogeneity assumptions. The Horvath and Watson test statistic in

14

the present context is simply the Wald statistic for testing the joint hypothesis "f = "s = 0 where the parameters are estimated by OLS from the VECM (11)11. Under the null of no-cointegration the Wald test has a nonstandard asymptotic distribution and Horvath and Watson (1995) supply the appropriate critical values. They show that their test can have considerably higher power than Johansen’s rank test, which is based on implicitly estimating the cointegrating vector. Additionally,they show that their test has good power even if the cointegrating vector is moderately misspecified.The estimates of the VECMs imposing the cointegrating vector (1,-1) are presented in Table 8 and the Horvath-Watson Wald statistics are reported in Table 5. Using the Horvath-Watson Wald test,the null of no-cointegration is rejected at the 5% level in favor of the alternative of cointegration with cointegrating vector (1,-1) for all three exchange rates.

Based on the results from Table 8, the FRUH is clearly rejected for all three exchange rate

series since the null hypothesis that "s = 1 can be rejected at any reasonable level of significance using an asymptotic t -test. As discussed in Engel (1996), a rejection of the FRUH is usually interpreted as evidence for the existence of a time varying risk premium or a peso problem. Using (11b), it is easy to see that the risk premium implied by the VECM (11) is rp t re = (1 - "s )(f t - s t ) and so the forward premium is perfectly correlated with the risk premium.

5. Models of Cointegration between s t+1 and f t implied by cointegration between

f t and s t

As discussed earlier, cointegration between f t and s t with cointegrating vector (1,-1) implies cointegration between s t+1 and f t with cointegrating vector (1,-1). However, the implied VECM and triangular representations for s t+1 and f t based on the simple models of cointegration between f t and s t presented in the previous section are somewhat nonstandard. To see this, consider first the derivation of the VECM for )f t and )s t+1. By adding and subtracting "s f t-1 from the right hand side of (11b), led one period, and adding and subtracting "f s t from the right hand side of (11a), we get the VECM for ()f t , )s t+1)N

)f t = μf - "f (s t - f t-1) + "f )s t + ,ft ,(14a))s t+1 = μs - "s (s t - f t-1) + "s )f t + ,s,t+1.

(14b)

Notice that in (14) the error correction term is now the lagged forecast error, s t - f t-1, and not the

15

lagged forward premium and that the error terms are separated by one time period and are thus contemporaneously uncorrelated. The errors, however, are not independent due to the correlation between ,ft and ,st . Consequently, the representation in (14) is not a VECM that can be derived from a finite order VAR model for (s t+1, f t )N . In addition, since the error correction term enters both equations neither the forward rate nor the future spot rate is weakly exogenous for the cointegration parameters.

Although )s t is on the right hand side of (14a) and )f t is on the right hand side of (14b) these models should not be interpreted as conditional models since they are derived by simple algebraic manipulation of (11). In particular, since the time index is shifted by one period between the two equations it is more appropriate to interpret )f t in (14b) as a predetermined variable. Estimation of (14b) by OLS will yield consistent but not necessarily efficient estimates of "s 12. However,estimation of (14a) by OLS is problematic since both s t - f t-1 and )s t are correlated with ,ft .

Next consider the derivation of the triangular representation. Using (8b) and f t - s t - μc = u ft

the triangular model for s t+1 and f t becomes

s t+1 = μc + f t +

(15b)

where

Now suppose that s t+1 and f t are cointegrated and (15a) is the correct representation given that $f = 1. Then the OLS estimate of $f from the levels regression (3) will be consistent but asymptotically biased and inefficient due to dynamic behavior and feedback between the elements of v t+1 = (v s,t+1, v ft )N . Since the VECM (14) cannot be derived from a finite order cointegrated VAR model for (s t+1 ,f t ), testing for cointegration and estimating the cointegrating vector using standard VAR techniques is problematic. In particular, since s t - f t-1 is correlated with ,ft , testing and estimation methods based on naive VECMs for )s t+1 and )f t , like the Horvath-Watson Wald test and the Johansen MLE, are likely to suffer from biases. The dynamic OLS/GLS estimator of Stock and Watson (1993), however, should work well since it is designed to pick-up feedback effects through the inclusion of leads and lags of )f t in the levels regression (3). The results of Table 10 show that this is indeed the case for the pound, yen and Canadian dollar. For all series, the OLS estimates of $f are downward biased but the Stock-Watson DOLS and DGLS estimates are nearly one.

Somewhat surprisingly, the triangular representation (15) shows that the OLS estimate of $f

will be biased even if "s = 1 (the FRUH is true). This result is due to the fact that the long-run covariance matrix of v t+1 is not diagonal. To illustrate, let the FRUH be true and suppose that "f =

17

S

'

F ss F F sf F '0

'

ss 00F ff

,'

'

0F 00

)

'

ss F 0

F ,

0 so that the forward premium is not autocorrelated (this assumption greatly simplifies the calculations but does not qualitatively affect the end result). Then the triangular representation (15)simplifies to

s t+1 = f t + ,s,t+1,f t = f t-1 + ,ft ,

and v t+1 = e t+1 = (,s,t+1, ,ft )N . Then by straightforward calculations the long-run covariance matrix of e t+1 and its components are

so that 2 = F sf /F ff and )fs = 0. Further, since F ss . F ff and D sf . 1 it follows that 2 . 1 and so OLS on the levels regression (3) will suffer from bias even if the FRUH is true. To illustrate the magnitude of the bias, Table 2(c) reports OLS estimates of the levels regression (3) when data are generated from (12). The OLS estimate of $f is biased downward, to a similar degree observed in empirical results, and the finite sample distribution is heavily left-skewed. The t -statistic for testing $f = 1 is centered around -1.5 and a nominal 5% test rejects the null that $f = 1 about 30% of the time when the null is true. Table 2 also reports results for the Stock-Watson DOLS estimator. In all cases, the Stock-Watson estimator is essentially equal to the true value of unity and the t -statistic for testing $f = 1 is roughly symmetric and centered around zero. However, there is moderate size distortion in the nominal 5% t -tests of $f = 1 for T =100 but the distortions dissipates as T increases.

6. A Reinterpretation of Some Recent Results Regarding the FRUH.

The results of the previous sections can be used to reinterpret the results of Norrbin and Reffett (1996), hereafter NR, and Naka and Whitney (1995), hereafter NW, who use particular cointegrated models for (s t+1, f t )N and find support for the FRUH. 6.1 Norrbin and Reffett’s model

NR based their analysis on the following VECM for (s t+1, f t )N

18

)s t+1 = μs + *s (s t - $f f t-1) + H st+1,(16a))f t = μf + *f (s t - $f f t-1) + H ft,

(16b)

which is based on a cointegrated VAR(1) model for (s t+1, f t )N 14. NR are primarily interested in directly testing the LRFRUC, i.e., that (s t+1, f t ) are cointegrated with cointegrating vector (1, -1), and not the FRUH. Their approach is to impose $f = 1, estimate (16) by OLS and test the significance of the error correction coefficients *s and *f . Table 11 presents the estimation results for (16) applied to our data. They find that estimates of *s are not statistically different from zero, estimates of *f are not statistically different from 1, the R2s from (16a) and (16b) are close to zero and one, respectively,and the error term from (16b) is highly serially correlated 15. Our results are very similar. From these results they conclude that s t+1 and f t are cointegrated with cointegrating vector (1,-1) (since *f … 0)and that spot rates are weakly exogenous for the cointegrating parameters (since *s = 0)16. Based on their finding that spot rates are weakly exogenous they argue that tests of the LRFRUC constructed from an error correction equation for )s t+1 are bound to lead one to mistakenly reject the LRFRUC and, therefore, reject the FRUH.

Using the cointegrated model for s t+1 and f t implied by the cointegrated model for s t and f t

presented in section 5 we can give alternative interpretations of NR’s results. Most importantly, our results show that NR’s claim that spot rates are weakly exogenous is inconsistent with the FRUH.To see how NR arrived at their results observe that (16) is a restricted version of (14) since )f t is omitted from (16a) and )s t is omitted from (16b). Now, NR’s finding that estimates of *s are close to zero can be explained by omitted variables bias. For example, if (14) is the true model, with μf =μs = 0, then straightforward calculations based on the stylized facts of the data show

and so plim *^

s . 0. Also, as

plim T &1

j T 1(s t &f t &1))f t .F 2

,plim T &1

j T 1

(s t &f t &1)2.F 2mentioned in the last section, estimation of (16b) by OLS is problematic due to the correlation between *f (s t - $f f t-1) and e ft . Furthermore, the finding that *6

f = 1 with $f = 1 in (16b) implies that )f t = s t - f t-1 + H ^ft or, equivalently, that f t = s t + H ^ft . This combined with the result that R2 = 1 and H ^ft is

highly autocorrelated simply shows that the forward premium is highly autocorrelated and does not provide evidence one way or another about the FRUH. Finally, consider NR’s Table 2 which gives the results for the estimation of the error correction model

)s t+1 = μ + "(s t - f t-1) + *)f t + B)f t-1 + ()s t + u t+1

(17)

19

which mimics the ECM estimated by Hakkio and Rush (1989). NR claim that this regression is misspecified since it mistakenly assumes that forward rates are weakly exogenous (presumably due to the presence of )f t ). However (17) is in the form of (14b) which is not a conditional model and does not make any assumptions about the weak exogeneity of forward rates so NR’s claim is not true.6.2 Naka and Whitney’s model

NW are interested in testing the LRFRUC and the FRUH simultaneously using the following cointegrated triangular representation for (s t+1, f t )N

s t+1 = μ + $f f t + v s,t+1,(18a))f t = v ft ,

(18b)

where

v s,t+1 = D v st + w s,t+1,(18c)v ft = w ft ,

(18d)and w s,t+1, w f t are i.i.d . error terms. Notice that (18a) allows for serial correlation in the “levels regression” but the restriction that f t is strictly exogenous is imposed in (18b). The VECM derived from (18) is

)s t+1 = (1 - D )μ - (1 - D )(s t - $f f t-1) + $f )f t + w s,t+1,(19a))f t = w ft .

(19b)

In (19a) the speed of adjustment coefficient is directly related to the correlation in the forecast error and the long-run impact of forward rates on future spot rates (the coefficient on f t ) is restricted to be equal to the short-run effect (the coefficient on )f t ) . In (19), the FRUH imposes the restrictions μ= 0, D = 0 and $f = 1. NW estimate (19a) by nonlinear least squares (NLS) and report estimates of μ and D close to zero and estimates of $f close to one 17. Table 12 replicates NW’s analysis using our data and we find very similar results. Based on these results, NW cannot reject the FRUH.

The triangular representation (18) used by NW is very similar to the triangular model (15) but with some important differences. In particular, since f t is assumed to be strictly exogenous and w s,t+1and w ft are assumed to be independent NW’s model does not allow for feedback between )s t and )f t or for contemporaneous correlation between

20

can be made. These assumptions, however, place unrealistic restrictions on the dynamics of spot and forward rates. For example, suppose (18) is the correct model and that the FRUH is true so μ = 0,$f = 1 and D = 0. Then the derived VECM for (f t , s t )N is

)f t = w ft

)s t = (f t-1 - s t-1) + w st

where w ft and w st are independent. This model implies that f t-1 - s t-1 = w f,t-1 - w s,t-1 , which is a white noise process, and that )f t and )s t are uncorrelated. Clearly these results are at odds with the observation that the forward premium is highly autocorrelated and that )f t and )s t are highly contemporaneously correlated. In addition, NW fail to recognize that assuming f t is strictly exogenous and w st and w ft are independent the results of Phillips and Loretan (1991) and Phillips (1991) show that OLS on (18a) yields asymptotically efficient estimates of $f and so there is no efficiency gain in estimating the nonlinear error correction model (19a)18. Indeed, the results of Tables 10 and 12 show that the OLS and NLS estimates of $f are almost identical. Finally, since NW’s estimates of $f are essentially unity, their tests for the significance of D in (19a) are roughly equivalent to tests for serial correlation in the forecast error s t+1 - f t . Given the remarks in section 5we know that this test of the FRUH is bound to have low power. In sum, by starting with a simple cointegrated model for s t+1 and f t , NW fail to capture some important dynamics between s t and f t that provide information about the validity of the FRUH.

7. Conclusion

In this paper we illustrate some potential pitfalls in modeling the cointegrated behavior of spot and forward exchange rates and we are able to give explanations for some puzzling results that commonly occur in exchange rate regressions used to test the FRUH. We find that a simple first order VECM for s t and f t captures the important stylized facts of typical monthly exchange rate data and serves as a natural statistical model for explaining exchange rate behavior. We show that the cointegrated model for s t+1 and f t derived from the VECM for s t and f t is not a simple finite order VECM and that estimating a first order VECM for s t+1 and f t can lead to mistaken inferences concerning the exogeneity of spot rates and the unbiasedness of forward rates.

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1.双击运行：辅助论坛Creo破解补丁.exe 2.指定目录如:PTC\Creo 1.0(2.0)\Common Files\Mxxx，然后破解 3.完成后会出现如下提示，点击[OK]，破解成功 26.破解Distributed Services： 1. 在右侧指定..\Creo 1.0( 2.0)\Creo Distributed Services Manager 后再次点击【look For】

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常见地遥感卫星地介绍及具体全参数

常见的遥感卫星的介绍及具体参数 遥感卫星(remote sensing satellite )用作外层空间遥感平台的人造卫星。用卫星作为平台的遥感技术称为卫星遥感。通常，遥感卫星可在轨道上运行数年。卫星轨道可根据需要来确定。遥感卫星能在规定的时间覆盖整个地球或指定的任何区域，当沿地球同步轨道运行时，它能连续地对地球表面某指定地域进行遥感。所有的遥感卫星都需要有遥感卫星地面站，卫星获得的图像数据通过无线电波传输到地面站，地面站发出指令以控制卫星运行和工作。以下列出较为常见的遥感卫星: 一、Landsat卫星 美国NASA的陆地卫星(Landsat)计划(1975年前称为地球资源技术卫星——ERTS )，从1972年7月23日以来，已发射7颗（第6颗发射失败）。目前Landsat1—4均相继失效，Landsat 5仍在超期运行（从1984年3月1日发射至今）。Landsat 7于1999年4月15日发射升空。其常见的遥感扫描影像类型有MMS影像、TM图像。 （一）、MSS影像 MSS影像为多光谱扫描仪（MultiSpectral Scanner）获取的图像，第一颗至第三颗地球卫星（Landsat）上反光束导管摄像机获取的三个波段摄影相片分别称为第1、2、3波段，多光谱扫描仪有4个波段获取的扫描影像被命名为4、5、6、7波段，两个波段为可见光波段，两个波段为近红外波段，此外，第三颗地球卫星上还供有热红外波段影像，这个影像称为第8波段，但使用不久，就因为一起的问题二关闭了。 表 1 ：Landsat上MSS波段参数

(二)、TM影像 TM影像是指美国陆地卫星4～5号专题制图仪（thematic mapper）所获取的多波段扫描影像。 影像空间分辨率除热红外波段为120米外，其余均为30米，像幅185×185公里2。每波段像元数达61662个（TM-6为15422个）。一景TM影像总信息量为230兆字节），约相当于MSS影像的7倍。 因TM影像具较高空间分辨率、波谱分辨率、极为丰富的信息量和较高定位精度，成为20世纪80年代中后期得到世界各国广泛应用的重要的地球资源与环境遥感数据源。能满足有关农、林、水、土、地质、地理、测绘、区域规划、环境监测等专题分析和编制1∶10万或更小比例尺专题图，修测比例尺地图的要求。 表 2 ：Landsat上TM波段参数 （三）、ETM 1999年4月15日，美国发射了Landsat-7，它采用了增强-加型专题绘图仪（ETM）遥感器来获取地球表层信息，它与TM的区别在于增加了全色波段，分辨率为15米，并改进了热红外波段影像的分辨率。

卫星全色和多光谱模式介绍

QuickBird卫星全色和多光谱模式 时间:2009-08-24 众所周知，遥感是使用各种传感器远距离探测目标所辐射、反射或散射的电磁波，经加工处理变成能够识别和分析的图像和信号，以获取目标性质和状态信息的综合技术。 遥感根据获取目标的手段不同可分为狭义遥感和广义遥感。 狭义遥感以电磁辐射为感测对象，而广义遥感还包括磁力、重力等地球物理的测量和属于地球物理测量范畴的地震波、声波等弹性波。 我们通常所说的遥感概念则专指以电磁辐射为特征的狭义遥感。不同的目标物受到太阳或其他辐射源的电磁辐射时，它们所特有的反射、发射、透射、吸收电磁辐射的性质是不同的。通过获取目标物对电磁辐射的显示特征，可识别目标的属性和状态。所以传感器谱段的设置与目标物的光谱特性有着密切的关系。 目前世界上用于卫星遥感的传感器有两大类：光学遥感和微波遥感。 光学遥感: 光学遥感指利用光学设备探测和记录被测物体辐射、反射和散射的相应谱段电磁波，并分析、研究其特性及变化的技术。 光学遥感覆盖了红外、可见光和紫外三个谱段，常用的有以下三种： 可见光遥感: 其工作波长为0.4～0.76微米，一般采用感光胶片或光电探测器作为感测元件，属于摄影成像遥感。它主要使用可见光远摄镜头照相和可变焦距电视摄像等，感测的是目标及背景反射或自身发出的可见光，记录的信息或拍摄的图像是物体反射光或发光强度的空间分布。可见光遥感是光学遥感中历史最长的一种，是对地观测和军事侦察的主要手段之一。摄影成像的分辨率(G)很高，可以近似地表示为： G=f×R/H 其中f为镜头焦距，R为镜头与底片的综合分辨率，H为高度(或距离)。 红外遥感器: 主要包括红外扫描仪、红外辐射仪等。红外遥感通过探测红外辐射获取目标和背景的辐射温度或热成像。其探测能力取决于目标、背景与周围环境的温度差。红外遥感的最大优点是可获取无光照或薄云下目标和背景的图像。 多谱段遥感： 使用几个不同的谱段同时对一目标或地区进行感测，从而获得与各谱段相对应的各种信息。将不同谱段的遥感信息加以组合，可获取目标物更多的信息。多谱段遥感是在可见光和红外遥感的基础上发展起来的，它能明显地分辨多种目标和背景特性，兼有可见光和红外遥感技术的优点。也为高光谱和超高光谱的发展提供了依据。微波遥感： 微波遥感是利用微波遥感设备，对地物目标和环境的微波辐射、反射或散射能量实施探测的技术，其波长为1～1000毫米． 微波遥感按工作模式的不同可分为两种： 有源微波遥感: 主要由成像雷达、微波散射计和微波高度计组成。在卫星遥感中应用较多的是合成孔径雷达，它是利用平台与目标的相对运动产生的多普勒频移，经二维相关处理或匹配滤波处理而获得高分辨率的图像。 无源微波遥感: 主要指各种微波辐射计，它是通过测量自然界各种物体发出的微弱微波辐射来测量目标的辐射特性和实际温度。

WorldView卫星影像命名规则

WorldView卫星影像命名规则 WorldView-2于2009年10月6日发射升空，运行在770Km高的太阳同步轨道上。更高的轨道带来了更短的重访周期和更好的拍摄机动性。作为Digital Globe公司当时先进的遥感卫星，它同样使用了控制力矩陀螺技术。这项高性能技术可以提供多达10倍以上的加速度的姿态控制操作，从而可以更精确的瞄准和扫描目标。卫星的旋转速度可从QuickBird的60秒减少至9秒，星下摆动距离达200km。所以，WorldView-2在太空中的角色就像一个神奇的画笔，能灵活的前后扫描、拍摄大面积的区域，能在单次操作中完成多频谱影像的扫描。除了更快速的采集和更高的精度，WorldView-2还是第一颗具有八波段多光谱的高分辨率遥感卫星，它不但具有传统遥感卫星的四个多光谱波段，还新增加了海岸线、黄、红边和近红外2波段。 一般情况下，我们订购的影像都是分块存储的，上图就是一幅分块影像的所有文件。 （1）*.ATT——姿态文件：存储第一个数据点的时间、数据点数目、点和姿态信息间隔。 （2）*.EPH——星历文件：存储第一个数据点获取的时间、数据点数目、点和星历信息之间的间隔。 （3）*.GEO——几何定标文件：虚拟相机的标注摄影测量参数，是基础产品的相机和光学系统之间的关系。

（4）*.IMD——影像元数据文件：存储影像关键信息，包括产品级别、角点坐标、投影信息、获取时间、分辨率、视线高度、方位角、云覆盖率等。对后期数据处理分析有很大帮助。 （5）*.RPB——RPC参数文件：包含影像的RPC参数，是影像物方空间坐标与像方空间坐标之间的数学映射。这是我们做卫星影像立体成图RPC空三的关键参数。 （6）*.STE——立体文件：包含构成立体的影像列表，重叠区域等。 （7）*.TIF——影像文件:原始影像格式为非标准16bit，普通看图软件无法打开显示，可将其转换成8bit后再打开。或者使用ArcGIS、ERdas等专业软件打开。 （8）*.TIL——影像分块文件：产品分块情况及各部分位置关系。 （9）*.XML——影像索引文件：包含索引、许可、影像元数据、分块、rpc 文件的索引信息。 （10）*README.TXT——高级影像索引文件：产品文件列表和辅助数据文件以及产品版本信息。 备注说明： 北京揽宇方圆200多颗遥感卫星数据资源，各卫星都有详细的价格体系表，不同行业根据自己遥感项目业务要求，对各卫星影像的分辨率、波段数量、质量以及影像拍摄的时间要求各异，而卫星影像的价格则主要由以上参数决定。 北京揽宇方圆信息技术有限公司是国内的领先遥感卫星数据机构，遥感行业的国家高新技术企业，整合全球200多颗遥感卫星数据资源，遥感卫星影像数据贯穿中国1960年至今的所有商业卫星影像数据，是中国遥感卫星数据资源最多的专业遥感卫星数据服务机构，提供多尺度、多分辨率、全覆盖的遥感卫

卫星遥感数据处理规范流程

北京揽宇方圆信息技术有限公司遥感卫星影像图像数据处理介绍 北京揽宇方圆信息技术有限公司是国内的领先遥感卫星数据机构，而且是整合全球的遥感卫星数据资源，分发不同性能、技术应用上可以互补的多种卫星影像，包括光学、雷达卫星影像、历史遥感影像等各种卫星数据服务，各种专业应用目的的图像处理、解译、顾问服务以及基于卫星影像的各种解决方案等。遥感卫星影像数据贯穿中国1960年至今的所有卫星影像数据，是中国遥感卫星数据资源最多的专业遥感卫星数据服务机构，提供多尺度、多分辨率、全覆盖的遥感卫星影像数据服务，最大限度的保证了遥感影像数据获取的及时性和完整性。 优势： 1：北京揽宇方圆国内老牌卫星数据公司，经营时间久，行业口碑相传，1800个行业用户选择的实力见证。 2：北京揽宇方圆遥感数据购买专人数据查询一对一服务，数据查询网址是卫星公司网。 3：北京揽宇方圆拥有大型正版遥感处理软件，遥感数据处理工程师有10年以上遥感处理工作经验，并有国家大型项目工作经验自主卫星数据处理软件著作权，最大限度保持遥感卫星影像处理的真实度。

4：北京揽宇方圆国家高新技术企业，通过ISO900认证的国际质量管理操作体系，无论是遥感卫星品质和遥感数据处理质量，都能得到保障。 5：影像数据官方渠道：所有的卫星数据都是卫星公司授权的原始数据，全球公众数据查询网址公开查询，影像数据质量一目了然，数据反应客观公正实事求是，数据处理技术团队国标规范操作，提供的是行业优质的专业化服务。 6：签定正规合同：影像数据服务付款前，买卖双方须签订服务合同，提供合同相应的正规发票，发票国家税网可以详细查询，有增值税普通发票和增值税专用发票两种发票类型可供选择。以最有效的法律手段来保障您的权益。 7：对公帐号转款：合同约定的对公帐号，与合同主体名发票上面的帐号名称一致，是由工商行政管理部门核准的公司银行账户，所有交易记录均能查询，保障资金安全。 8：售后服务：完善的售后服务体制，全国热线，登陆官网客服服务同步。 技术能力说明 北京揽宇方圆拥有大型正版遥感处理软件，遥感数据处理工程师有10年以上遥感处理工作经验，并有国家大型项目工作经验自主卫星数据处理软件著作权，最大限度保持遥感卫星影像处理的真实度。 一．图像预处理 1．降噪处理 由于传感器的因素，一些获取的遥感图像中，会出现周期性的噪声，我们必须对其进行消除或减弱方可使用。 （1）除周期性噪声和尖锐性噪声 周期性噪声一般重叠在原图像上，成为周期性的干涉图形，具有不同的幅度、频率、和相位。它形成一系列的尖峰或者亮斑，代表在某些空间频率位置最为突出。一般可以用带通或者槽形滤波的方法来消除。 消除尖峰噪声，特别是与扫描方向不平行的，一般用傅立叶变换进行滤波处理的方法比较方便。