风险管理hansen model作业
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风险管理作业----Hansen's Basic Model1、给定校准值：利用稳态方程组，解出稳态值，并且校准参数A 。
对比书上结果（P108）欧拉方程：约束条件：随机过程：稳态方程组：校准： People spent a third of their time working, so H=1/3.MATLAB 求解方程组：alpha=0.36;beta=0.99;delta=0.025;H=1/3; syms A C K Y;S1=beta*(alpha*(Y/K)+1-delta)-1; S2=(1-H)*(1-alpha)*Y-A*C*H; S3=(K^alpha)*(H^(1-alpha))-Y; S4=Y-C-delta*K;[C,A,K,Y]=vpasolve(S1,S2,S3,S4,C,A,K,Y)C =0.918109157712174691897910466710990.99β=0.025δ=0.36α=111=[(1)]t t t t C E r C βδ++⋅+- (1)(1)tt t tY A C H H α⋅=-- 1 t t t t Y K H ααλ-=⋅⋅1(1)t t t t K K Y C δ+=-+-tttY r K α=11t t t λρλε++=⋅+1=(1)YK αβδ⋅+- (1)(1)A CH H Y α⋅=-- 1 Y K H αα-=⋅K Y C δ=-0.025δ=0.99β=0.36α=A =1.7213622291021671826625386996904 K =12.663084512717455889799353341169 Y =1.2346862705301110891428943002402校准后得到A=1.72，再把A 带回方程组校准H 。
alpha=0.36;beta=0.99;delta=0.025;A=1.72; syms H C K Y;S1=beta*(alpha*(Y/K)+1-delta)-1; S2=(1-H)*(1-alpha)*Y-A*C*H; S3=(K^alpha)*(H^(1-alpha))-Y; S4=Y-C-delta*K;[C,H,K,Y]=vpasolve(S1,S2,S3,S4,C,H,K,Y)C =0.91859378752813681894435247137647 H =0.33350928547435097653438264791977 K =12.669768803213156434665914117592 Y =1.2353380076084657298110003243163P108R=vpa(alpha*Y/K)R =0.03510101010101010101010101010101 H=0.3335，符合实际情况，运算结果和书上一致。
2、给定 利用公式计算均衡运动方程组，对比书上结果（P109）。
均衡运动方程组：0.95,ρ=1t kk t k tK v K v λλ+=⋅+⋅t yk t y t Y v K v λλ=⋅+⋅t ck t c t C v K v λλ=⋅+⋅t hk t h tH v K v λλ=⋅+⋅简化后的均衡运动方程组：其中，方程组系数为：；；代入求解过程：t rk t r tr v K v λλ=⋅+⋅kkP v =k Q v λ=yk ck hk rk v v R v v ⎛⎫ ⎪ ⎪= ⎪ ⎪ ⎪⎝⎭y c h r v v S v v λλλλ⎛⎫ ⎪ ⎪= ⎪ ⎪ ⎪⎝⎭1t t t x x z P Q -=⋅+⋅1t t t y x z R S -=⋅+⋅1t t x K +=1t t x K -=t tz λ=t t t t t Y C y H r ⎛⎫⎪ ⎪= ⎪ ⎪ ⎪⎝⎭求解系数：MATLAB 求解： A=[0 -K 0 0]';B=[0;((1-delta)*K);alpha;-1];C=[1 -1 (-1/(1-H)) 0;Y -C 0 0;-1 0 (1-alpha) 0;1 0 0 -1]; D=[0 0 1 0]'; F=[0]; G=F; H=F;J=[0 -1 0 (beta*R)]; K=[0 1 0 0]; L=F;M=F; N=[0.95]; syms P Q R S;S5=(F-J*(C^(-1))*A)*(P^2)-(J*(C^(-1))*B-G+K*(C^(-1))*A)*P-K*(C^(-1))*B+H; P=vpasolve(S5,P) P =0.953673888224406081000881937424320F G H ===(0,1,0,)J r β=- (0,1,0,0)K = 0L M ==N ρ=,01P P →<<求根公式 121110()()F JC A JC B G KC A B P K P C H ----=-⋅--+⋅-+1()R C AP B -=-⋅+1111[()()]()N F JC A JR FP G KC A JC D L N KC QD M-----+++-⋅=-+- 1()S C AQ D -=-⋅+Q→1.059168152314270270581541201087P=0.95367388822440608100088193742432R=(-C^(-1))*(A*P+B)R =0.204460190600975842612215326468470.56910285836708846048922735169117-0.24303095218597524591841355239302-0.79553980939902415738778467353153S6=(J*(C^(-1))*D-L)*N+K*(C^(-1))*D-M-(N*(F-J*(C^(-1))*A)+(J*R+F*P+G-K*( C^(-1))*A))*Q;Q=vpasolve(S6,Q)Q =0.11318305485357126424173303507069S=(-C^(-1))*(A*Q+D)S =1.45228269438704096504437144819550.391965281739043148588941723126880.706691709979751507881830387805461.4522826943870409650443714481955总结得：P=0.9537(P的绝对值小于1)，Q=0.1132R=[0.2045 0.5691 -0.2430 -0.7995]’，S=[1.4523 0.3920 0.7067 1.4523]’.稳态方程为：K t+1=0.9537Kt+0.1132λt，Yt=0.2045Kt+1.4523λt，Ct=0.5691Kt+0.3920λt，Ht=-0.2430Kt+0.7067λt，Rt=-0.7955Kt+1.4523λt.与书上结果一致。
3、给定我国年度校准值：（1）解出稳态值，并校准参数A 。
把H=1/3代入方程组；MATLAB 求解方程组：alpha=0.48;beta=0.98;delta=0.1;H=1/3; syms C A K Y;S1=beta*(alpha*(Y/K)+1-delta)-1; S2=(1-H)*(1-alpha)*Y-A*C*H; S3=(K^alpha)*(H^(1-alpha))-Y; S4=Y-C-delta*K;[C,A,K,Y]=vpasolve(S1,S2,S3,S4,C,A,K,Y)C =0.71845038341438257299527353542045 A =1.7294250281848928974069898534386 K =4.7626699599510366732944852178943 Y =1.1947173794094862403247220572099调试后得到A 的校准值为1.73，再把A 带回原方程组校准H 。
alpha=0.48;beta=0.98;delta=0.1;A=1.73; syms C H K Y;S1=beta*(alpha*(Y/K)+1-delta)-1; S2=(1-H)*(1-alpha)*Y-A*C*H; S3=(K^alpha)*(H^(1-alpha))-Y; S4=Y-C-delta*K;[C,H,K,Y]=vpasolve(S1,S2,S3,S4,C,H,K,Y)C =0.71829117937201863342206144993360.98β=0.1δ=0.48α=H =0.33325946878353776433732636399581 K =4.7616145825337875586490657560423 Y =1.1944526376253973892869680255378 H=0.3332，符合实际。
（2）根据我国实际年度数据，利用Solow 残差校准1、将我国生产总值，资本存量，就业人口数据取对并滤波。
2、solow 残差序列：对{lnSRt}序列线性滤波。
使用EVIEWS 操作，结果如下，得出=0.6109=0.01548（3）计算均衡运动方程组，模拟产出序列，校准随机项标准差 MATLAB 求解均衡运动方程： A=[0 -K 0 0]';B=[0;((1-delta)*K);alpha;-1];C=[1 -1 (-1/(1-H)) 0;Y -C 0 0;-1 0 (1-alpha) 0;1 0 0 -1]; D=[0 0 1 0]'; F=[0];,ρln (1)ln l ln n t t t t Y H K SR αα=--⋅-⋅11t t t λρλμ++=⋅+G=F;H=F;J=[0 -1 0 (beta*R)];K=[0 1 0 0];L=F;M=F;N=[0.61];syms P Q R S;S5=(F-J*(C^(-1))*A)*(P^2)-(J*(C^(-1))*B-G+K*(C^(-1))*A)*P-K*(C^(-1))*B+H; P=vpasolve(S5,P)P =0.900178588346542003250469450025621.1335619136860421590829738773955P=0.90017858834654200325046945002562;R=(-C^(-1))*(A*P+B)R =0.390644392398002363044699838807820.6484220736295228183589528336172-0.17183770692691853260634646383112-0.60935560760199763695530016119218S6=(J*(C^(-1))*D-L)*N+K*(C^(-1))*D-M-(N*(F-J*(C^(-1))*A)+(J*R+F*P+G-K*( C^(-1))*A))*Q;Q=vpasolve(S6,Q)Q =0.29901680666901196357951407281587S=(-C^(-1))*(A*Q+D)S =1.37186831312690714431022533688910.299083358115442719289785370635990.715131371397898354442741032479021.3718683131269071443102253368891得到：P=0.9002Q=0.2990R=[ 0.3906 0.6484 -0.1718 -0.6093]’S=[1.3719 0.2991 0.7151 1.3719]’得到均衡运动方程：K t+1=0.9002Kt+0.2990λt，Yt=0.3960Kt+1.3719λt，Ct=0.6484Kt+0.2991λt，Ht=-0.1718Kt+0.7151λt，Rt=-0.6093Kt+1.3719λt.校准随机项标准差：clear allrho=0.611;P=0.9002;Q=0.2990;R=0.3906; S=1.3719;sgm=0.0148; X=zeros(1001,1);Y=zeros(1001,1);Z=zeros(1001,1);U=rand(1001,1)*2*3^(1/2)*sgm-3^(1/2)*sgm;for i=2:1001Z(i,1)=rho*Z(i-1,1)+U(i,1)X(i,1)=P*X(i-1,1)+Q*Z(i,1)Y(i,1)=R*X(i-1,1)+S*Z(i,1)endY(1,:)=[]STDEVY=std(Y)标准差0.0277 实际的产出的标准0.0134 需要校准调试出sgm=0.0065（4）利用得到的均衡运动方程组以及校准后的随机项标准差，对各个变量进行模拟，计算模拟序列的描述统计量clear allrho=0.611;P=0.9002;Q=0.2990;R=0.3906; S=1.3719;sgm=0.0065;R1=0.3906;R2=0.6484 ;R3=-0.1718;R4=-0.6093;S1=1.3719 ;S2=0.2991 ;S3=0.7151 ;S4=1.3719;z=zeros(1001,1); x=zeros(1001,1) ;Y=zeros(1001,1);H=zeros(1001,1) ;C=zeros(1001,1);z=zeros(1001,1); U=rand(1001,1)*2*3^(1/2)*sgm-3^(1/2)*sgm;for i=2:1001z(i,1)=rho*z(i-1,1)+U(i,1)x(i,1)=P*x(i-1,1)+Q*z(i,1)Y(i,1)=R1*x(i-1,1)+S1*z(i,1)C(i,1)=R2*x(i-1,1)+S2*z(i,1)H(i,1)=R3*x(i-1,1)+S3*z(i,1)endx(1,:)=[];STDEVX=std(x)Y(1,:)=[];STDEVY=std(Y)C(1,:)=[];STDEVC=std(C)H(1,:)=[];STDEVH=std(H)STDEVX =0.0115STDEVY =0.0139 STDEVC =0.0087 STDEVH =0.0056（5）计算实际数据的相关描述统计量，进行静态的模型有效性检验 计算模拟序列{It}，It=(-)/(-)模拟数据的有效性检验SD SD/SD(Y)Autocor. Cor.With YY 0.0139 1 0.77786 1 C 0.0087 0.60448 0.97234 0.81370 H 0.0056 0.507463 0.57699 0.80036 I 0.07465.3669060.594910.86206实际数据的有效性检验SDSD/SD(Y)Autocor. Cor.With YY 0.0132949310.703157 1 C 0.0126610860.9523243820.692561 0.993002 H 0.0216210541.6262630820.610943 -0.290410 I 0.0256055291.9259619270.7678570.805533t t tY C I =+Y C I =+t t t Y Y C C I I ⋅=⋅+⋅{}t I →。
