Economics Computation Winter Prof. Garey Ramey. Exercises : 5 ; B = AX = B: 5 ; h 3 1 6

Transcription

1 Economics 80 - Computation Winter Prof. Garey Ramey Exercises Exercise 1. Consider the following maices: 9 A = 1 8 ; B = : a. Enter A and B as Matlab variables. b. Calculate the following maices: AB; A ; A 0 B 1 ; b ij ; [aij =b ij ] : c. Calculate the solution of the following linear system: AX = B: d. If a ij + b ij, set c ij = a ij, and otherwise set c ij = 0. e. Calculate the following maices: ; ; h 1 i : Exercise. Consider the AR(1) process y t = :1 + :y t 1 + " t ; 1

2 where the " t s are i.i.d.normal random variables with mean 0 and variance :0. a. Compute the mean of this process. Also compute the autocovariances for j = 0; :::; ;. b. Create a Matlab script with the name "Ex_Solution". In the script, draw T = 100 values " 1 ; :::; " T from a normal disibution using the function randn, and then consuct y 1 ; :::; y T using the above equation, with y 0 equal to the mean of the process. Calculate the sample mean and autocovariance function of the simulated data: Sample mean = ^ = 1 T Sample autocovariance = ^ j = 1 T c. Repeat part b with T = 10; 000. j X T X T t=1 y t; t=j+1 (y t ^)(y t j ^): Exercise. Consider the AR() process y t = : + :y t 1 + :y t + " t ; where the " t s are i.i.d.normal random variables with mean 0 and variance :0. a. Show that the process is stationary. b. Calculate the coe cients 1 ; :::; of the MA(1) representation. c. Suppose " t = 1 and " s = 0 for all s = t. Use the MA(1) representation to calculate the implied path y t ; y t+1 ; :::; y t+. Exercise.a. Copy the le US_Data.xlsx into your current Matlab folder. Use the command xlsread to create the variables Real GDP and Hours as Matlab vectors. b. Plot the log levels of the two series. c. Consider the following linear end model of log of real GDP: y t = + t + yt c ;

3 where t is the time index of quarters (i.e., t = 1 for 198-I, t = for 198-II, etc.). Estimate and by regressing y t on a constant term and vector of time indices. d. Use the estimated coe cients to calculate the end component: t = ^ + ^t: Plot y t and t together. e. Calculate and plot the cyclical component y c t = y t t. Calculate the standard deviation and rst-order autocorrelation of y c t. Exercise. f t g T t=1 For a given series fy t g T t=1, the HP lter determines the end component as the solution to min f t gt t=1 XT (y t t=1 t ) + X T 1 t= ((y t+1 yt ) Necessary and su cient conditions for a solution are given by (1 + ) 1 + = y 1 ; ( t 1 + (1 + ) + = y ; yt 1)) : yt yt 1 + (1 + )yt t+1 + t+ = y t ; t = ; :::; T ; yt yt yt yt + (1 + ) T 1 T = y T 1 ; 1 + (1 + ) T = y T : For T =, these conditions may be expressed as = y 1 y y y y y y : (1)

4 Let H denote the maix in (1). a. Write a Malab script that calculates the maix H for arbiary and T using the for, if, elseif and else commands. b. Use your script to create a Matlab function that calculates the end component for a given series y = fy t g T t=1 and smoothing parameter. c. Using the data from the le US_Data.xlsx, calculate the end component of the log of Real GDP for = 100 and = 10. Generate a plot of the log of Real GDP along with these two end components. d. Calculate and plot the cyclical components y c t = y t t for = 100 and = 10. Calculate the standard deviation and rst-order autocorrelation for each case. Exercise. Consider the following vector autoregression model of real GDP and hours: y t h t = cy c h + y y y h h y h h y t 1 h t 1 where " y t and "h t are uncorrelated white noise processes. + y h t + "y t " h t ; () a. Import the variables Real GDP and Hours from the le US_Data.xlsx. Take logs of these series and use them series to estimate () using ordinary least squares. b. Calculated the tted residuals ^" y t and ^" h t. Plot the two series, and calculate their standard deviations (denoted by ^ y and ^ h ) and rst-order autocorrelations. c. Consider the following model of the cyclical components of real GDP and hours: yc t h c t = ^ y y ^ h y ^ y h ^ h h yc t 1 h c t 1 + y t h t ; () where ^ y y, ^ y h, ^ h y and ^ h h are the estimates from part b, and y t and h t are normally disibuted random variables with zero means and standard deviations ^ y and ^ h, respectively.

5 Draw N = 1000 samples of the random vectors y 1 y. ; h 1 h. ; y T h T where T = 0. For each sample, calculate the paths of yt c and h c t implied by () for t = 1; :::; T, where y0 c = hc 0 = 0. d. For each of the N replications, calculate the standard deviations of the yt c and h c t paths. Also calculate the standard deviation of the standard deviations across the N replications for each of the variables. Exercise. The social planner solution for a common speci cation of the Real Business Cycle (RBC) model is determined by: X 1 max E 0 fc t;h t;k t+1 g 1 t=1 t (ln C t + ln(1 H t )) ; t=1 subject to A t Kt 1 Ht + (1 )K t = C t + (1 + g)k t+1 ; t = 1; ; :::; () A t+1 = A t e" t+1 ; t = 1; ; :::; () where f" t g is an exogenous white noise process having standard deviation, and K 0 and A 0 are given. The parameters satisfy ; ; ; (0; 1) and ; g; > 0. Necessary conditions for a solution are along with () and (). C t 1 H t = A t K 1 t H 1 t ; t = 1; ; :::; () E t (1 + g) 1 C t C t+1 (1 )At+1 K t+1 H t = 1; t = 1; ; :::; ()

6 The deterministic steady state equilibrium (DSSE) is the iple f C; H; Kg such that fc t ; H t ; K t+1 g = f C; H; Kg for all t satis es (), () and (). This may be expressed as C 1 H = K 1 H 1 ; (8) (1 + g) 1 (1 ) K H a + 1 = 1; (9) K 1 H a + (1 ) K = C + (1 + g) K: (10) a. Use fsolve to calculate the DSSE under the following parameter values: g : :98 :0 :8 :00 : b. Output in the DSSE is given by Y = K 1 H a : Create a Matlab function that calculates Y as a function of the ve parameters. Use the function to analyze how Y is a ected by increases in the parameters. Exercise 8. For each period t = 1; ; :::, a rm receives s t orders, and hires a t workers to service the orders at a rate of one order per worker. The rm pays a wage of w per worker per period, and in addition incurs an adjustment costs of (a t a t 1 ). Pro ts in period t are given by The rm s objective function is and the initial condition is a 0 = 0. a. Formulate the rm s pro t maximization problem as a dynamic programming problem. t = minfs t ; a t g wa t (a t a t 1 ) : E P 1 t=1 t t j s 1 ;

7 b. Suppose the s t s are i.i.d. uniform over the integers 0 through 19. Let the other parameter values be given by w : :1 :9 : Use Matlab to calculate the optimal policy function a(a t 1 ; s t ) using the grids S = 0 : 19 and A = 0 : 19. c. Calculate and plot the optimal numbers of workers a 1; :::; a 10 under the assumption s 1 = s 10 = 10. d. Suppose instead that adjustment costs are given by (minfa t a t 1 ; 0g). Repeat your calculations for parts b and c under this assumption.