TUTORIAL: BASIC OPERATIONS

Transcription

1 TUTORIAL: BASIC OPERATIONS [1] AN EXERCISE ON HOW TO USE GAUSS On your computer, create a file folder called seminar. And download the two files called exer.txt and ols.prg from Dr. Ahn s website. Locate the two files in the folder seminar. Then, follow the steps described below. The pictures shown below would be slightly different from what you will actually see from your screens. 1. Double click on the GAUSS 6.0 icon. Then, you will see: 2. Type chdir c:\seminar (or a: if you use a floppy disk) on the command window and push the return button. By doing so, you can change your working directory to seminar Basic-1

2 3. Click on file/open. Then, you will see: Click on the Open button. Then, Basic-2

3 You can edit the program if you want. Once you finish editing, save the program using the save button on the screen. If you would like to create your own program, on command window, type, edit johndoe.prg (or other name) and push the return key. Then, you can get an empty window named johndoe.prg. Type your own codes on it! 4. To run the program, click on Action\Run Active File. To see the output file, click on file/open. When the list of the files appears, click on ols.out, which is an output file created by running ols.prg. Then, you will see: Basic-3

4 Basic-4

5 [2] SOME USEFUL COMMANDS (1) Reading data from a text file: The form of exer.txt: This data set contains 100 observations for 5 variables. To read this data set: load data[100,5] = exer.txt; y = all the entries on the first x2 = all the entries on the second x3 = data[.,3]; x4 = data[.,4]; x5 = data[.,5]; The last 5 lines define variable names. Basic-5

6 (2) Making Comments GAUSS does not execute any codes or between /* and */. Thus, you can make some comments in the or /*... */. (3) Matrix Operation Let A = [a ij ] and B = [b ij ] be conformable matrices. A*B: Product of A and B. A : Transpose of A. A B: Product of A and B. A[.,1]: The first column of A. A[1,.]: The first row of A. A[1,2]: The (1,2)th element of A. A[1:5,.]: The 1st, 2nd, 3rd, 4th and 5th rows of A. A[1 3,.]: The 1st and 3rd rows of A. inv(a): Inverse of A. invpd(a): Inverse of a positive definite matrix A. diag(a): n 1 vector of diagonal elements of an n n matrix A. sqrt(a): [ a ij ]. A^2: [a ij 2 ]. A B: Merge A and B vertically. A~B: Merge A and B horizontally. A.*.B: Kronecker Product of A and B. sumc(a): n 1 vector of sums of individual columns for an m n matrix A. 1 2 EX: 9 A = 3 4 sumc( A) = meanc(a): n 1 vector of means of individual columns for a m n matrix A. 1 2 EX: 3 A = 3 4 meanc( A) = stdc(a): n 1 vector of standard errors of individual columns for an mn matrix A. Suppose A = [a ij ] m n ; B = [b ij ] m n ; C = [c ij ] m 1 ; and d = scalar. A./B: [a ij /b ij ] (element by element operation). Basic-6

7 A./C: [a ij /c i ] (element by element operation). d-a: [d-a ij ]. d*a: [da ij ]. A/d: [a ij /d]. Generating a matrix of standard normally distributed random variables: rndns(t,k,dd): random matrix of t rows and k columns generated using dd as a seed number seed number. For uniformly distributed (between zero and 1) random variables use rndus[t,k,dd]. (4) Do Loop When you need to do the same job repeatedly, the following command would be useful: i = 1; do while i <= 100; : : : i = i + 1; endo; If you run the above commands, then the codes written on (: : :) are repeatedly executed. (5) Drawing Histogram Run the following command lines: library pgraph; graphset; {a1,a2,a3} = hist(storb, 50); If you execute the three lines, GAUSS draw a histogram for the variable named storb counting the frequencies for each of 50 categories. The variables named a1, a2 and a3 will contain some numeric information about the histogram (see the Gauss manual for detail). Basic-7

8 Or, you can run the following command lines: library pgraph; graphset; v = seqa(0,0.01,200); {a1,a2,a3} = hist(storb,v); The v is a vector of an additive sequence. The first argument in sega is the starting value. The second argument is increment, and the third argument is the number of elements in the sequence. Thus, the vector v is in the form of (0, 0.1, 0.2,..., 1.99). If you execute the above four lines, GAUSSLIGHT draw a histogram for the variable named storb counting the frequencies for the entries of v as breakpoints to be used to compute the frequencies. Basic-8

9 [3] OLS EXERCISE Program file: ols.prg /* ** OLS Program Data load data[100,5] = Define # of observations, # of regressors, X and tt = # of kk = # of y = data[.,1]; x2 = data[.,2]; x3 = data[.,3]; x4 = data[.,4]; x5 = data[.,5]; vny = name of the Dependent yy = y ; vny = xx = ones(tt,1)~x2~x3~x4~x5; vnx = {"cons", "x2", "x3", "x4", "x5" } Do not OLS using yy and b = invpd(xx'xx)*(xx'yy); e = y - xx*b ; s2 = (e'e)/(tt-kk); v = s2*invpd(xx'xx); econ = b~sqrt(diag(v))~(b./sqrt(diag(v))); econ = vnx~econ; Basic-9

10 sst = yy'yy - tt*meanc(yy)^2; sse = e'e; r2 = 1 - Printing out OLS output file = ols.out reset; let mask[1,4] = ; let fmt[4,3] = "-*.*s" 8 8 "*.*lf" 10 4 "*.*lf" 10 4 "*.*lf" 10 4; format /rd 10,4 ; "" ; "OLS Regression Result" ; " " ; " dependent variable: " $vny ; "" ; " R-Squares " r2 ; "" ; "variable coeff. std. err. t-st " ; yyprin = printfm(econ,mask,fmt); "" ; output off ; Output file: ols.out OLS Regression Result dependent variable: y R-Squares variable coeff. std. err. t-st cons x x x x [4] OLS MONTE CARLO EXPERIMENTS Basic-10

11 Theory says that OLS estimators are normally distributed under ideal conditions. What does it mean? Consider the data exer.txt. The data is generated by the following equation: y t = 0 + x t2 + 2x t3 + 3x t4 + 4x t5 + ε t, t = 1, , and var(ε t ) = 4. Observe that β 2 = 1. We now generate a new data set (say, Data Set 1) as follows: 1. For each t = 1,..., 100, keep the values of x t2,..., x t5, but draw ε t from N(0,4). Then, generate new y t following the above equation. 2. Estimate β2. Name this estimate [1] β2. 3. Generate other data set (say, Data Set 2) by the same way you generate Data Set 1. Then, get β. [2] 2 [1] 4. Repeat 5000 (or more) times and get [5000] β2,..., β2. The econometric theory implies that the estimates will be normally distributed. Let s check whether it is true or not. Program file: olsmonte.prg /* ** Monte Carlo Program Load load data[100,5] = exer.txt Data generation under Classical Linear Regression seed = 1; tt = # of kk = # of iter = # of sets of different xx = Regressors are tb = {0,1,2,3,4} y = x(2)*1 + x(3)*2 + x(4)*3 + x(5)*4 + storb = zeros(iter,1); storse = zeros(iter,1); i = 1; do while i <= Generating Basic-11

12 yy = xx*tb + OLS using yy and b = invpd(xx'xx)*(xx'yy); e = yy - xx*b ; s2 = (e'e)/(tt-kk); v = s2*invpd(xx'xx); se = sqrt(diag(v)); storb[i,1] = b[2,1]; storse[i,1] = se[2,1]; i = i + 1; Reporting Monte Carlo output file = olsmonte.out reset; format /rd 12,3; "Monte Carlo results"; " "; "Mean of OLS b(2) =" meanc(storb); "s.e. of OLS b(2) =" stdc(storb); "mean of estimated s.e. of OLS b(2) =" meanc(storse) ; library pgraph; graphset; {a1,a2,a3}=hist(storb,50); output off ; Output file: olsmonte.out Monte Carlo results Mean of OLS b(2) = s.e. of OLS b(2) = mean of estimated s.e. of OLS b(2) = Basic-12

13 Graphic Outcome: Basic-13

14 [5] OLS Under Heteroskedasticity Program file: hetmonte.prg /* ** Monte Carlo Program for OLS, OLS with White Correction ** and GLS under Heteroskedasticity Data generating seed = seed tt = # of kk = # of iter = # of sets of different x = Nonstochastic tb = y = x(1)*1 + x(2)*2 + rho = 0.1 degree of storob = zeros(iter,1); storose = zeros(iter,1); storosec = zeros(iter,1); storot = zeros(iter,1); storotc = zeros(iter,1); storgb = zeros(iter,1); storgse = zeros(iter,1); storgt = zeros(iter,1); storfb = zeros(iter,1); storfse = zeros(iter,1); storft = zeros(iter,1); i = 1; do while i <= Generating y with y = x*tb + OLS using y and b = invpd(x'x)*(x'y); e = y - x*b ; ee = e^2; s2 = (e'e)/(tt-kk); v = s2*invpd(x'x); vc = (x.*e)'(x.*e); vc = invpd(x'x)*vc*invpd(x'x); Basic-14

15 se = sqrt(diag(v)) ; sec = sqrt(diag(vc)) ; storob[i,1] = b[2,1]; storose[i,1] = se[2,1]; storosec[i,1] = sec[2,1]; olst = (b[2,1]-tb[2,1])/se[2,1]; df = tt-2; pval = 2*cdftc(abs(olst),df); if pval >= 0.05; storot[i,1]= 0; else; storot[i,1] = 1; endif; olstc = (b[2,1]-tb[2,1])/sec[2,1]; df = tt-2; pval = 2*cdftc(abs(olstc),df); if pval >= 0.05; storotc[i,1]= 0; else; storotc[i,1] = 1; infeasible GLS using y and ys = y./sqrt(rho*x[.,2]+(1-rho)*1); xs = x./sqrt(rho*x[.,2]+(1-rho)*1); b = invpd(xs'xs)*(xs'ys); e = ys - xs*b ; s2 = (e'e)/(tt-kk); v = s2*invpd(xs'xs); se = sqrt(diag(v)) ; storgb[i,1] = b[2,1]; storgse[i,1] = se[2,1]; glst = (b[2,1]-tb[2,1])/se[2,1]; df = tt-2; pval = 2*cdftc(abs(glst),df); if pval >= 0.05; storgt[i,1]= 0; else; storgt[i,1] = 1; feasible GLS using y and fh = x*invpd(x'x)*(x'ee); ys = y./sqrt(abs(fh)); xs = x./sqrt(abs(fh)); b = invpd(xs'xs)*(xs'ys); e = ys - xs*b ; Basic-15

16 s2 = (e'e)/(tt-kk); v = s2*invpd(xs'xs); se = sqrt(diag(v)) ; storfb[i,1] = b[2,1]; storfse[i,1] = se[2,1]; glst = (b[2,1]-tb[2,1])/se[2,1]; df = tt-2; pval = 2*cdftc(abs(glst),df); if pval >= 0.05; storft[i,1]= 0; else; storft[i,1] = 1; endif; i = i + 1; Reporting Monte Carlo output file = hetmonte.out reset; format /rd 12,3; "Monte Carlo results"; "" ; " "; "Checking Unbiasedness"; "-----"; "Mean of OLS b(2) =" meanc(storob); "s.e. of OLS b(2) =" stdc(storob); "mean of estimated s.e. of OLS b(2) =" meanc(storose) ; "mean fo White-corrected s.e. of OLS b(2) =" meanc(storosec) ; "-----"; "Mean of infeasible GLS b(2) =" meanc(storgb); "s.e. of infeasible GLS b(2) =" stdc(storgb); "mean of estimated s.e. of infeasible GLS b(2) =" meanc(storgse) ; "-----"; "Mean of feasible GLS b(2) =" meanc(storfb); "s.e. of feasible GLS b(2) =" stdc(storfb); "mean of estimated s.e. of feasible GLS b(2) =" meanc(storfse) ; ""; "Checking sizes of t-tests at 5% sig. level"; "-----"; "Rejection Rate of the t-test with usual OLS =" meanc(storot); "Rejection Rate of the t-test with White-corrected OLS =" meanc(storotc); "Rejection Rate of the t-test with infeasible GLS =" meanc(storgt); "Rejection Rate of the t-test with feasible GLS =" meanc(storft); Basic-16

17 output off ; Output file: hetmonte.out Monte Carlo results Checking Unbiasedness Mean of OLS b(2) = s.e. of OLS b(2) = mean of estimated s.e. of OLS b(2) = mean fo White-corrected s.e. of OLS b(2) = Mean of infeasible GLS b(2) = s.e. of infeasible GLS b(2) = mean of estimated s.e. of infeasible GLS b(2) = Mean of feasible GLS b(2) = s.e. of feasible GLS b(2) = mean of estimated s.e. of feasible GLS b(2) = Checking sizes of t-tests at 5% sig. level Rejection Rate of the t-test with usual OLS = Rejection Rate of the t-test with White-corrected OLS = Rejection Rate of the t-test with infeasible GLS = Rejection Rate of the t-test with feasible GLS = Basic-17

18 [5] OLS Under Autocorrelation Program file: automonte.prg /* ** Monte Carlo Program ** for OLS and GLS ** under Autocorrelation: AR(1) */ seed = 1; tt = # of kk = # of iter = # of sets of different x = Nonstochastic tb = {1,2} y = x(1)*1 + x(2)*2 + rho =.9 e(t) = rho*e(t-1) + vvar = 4; bandw = 5; storb = zeros(iter,1); storse = zeros(iter,1); storsec = zeros(iter,1); stort = zeros(iter,1); stortc = zeros(iter,1); storpwgb = zeros(iter,1); storpwgse = zeros(iter,1); storpwgsec = zeros(iter,1); storpwgt = zeros(iter,1); storcogb = zeros(iter,1); storcogse = zeros(iter,1); storcogsec = zeros(iter,1); storcogt = zeros(iter,1); i = 1; do while i <= Generating y with ep= sqrt(vvar/(1-rho^2))*rndns(tt,1,seed); j = 2; do while j <= tt; ep[j,1] = rho*ep[j-1,1]+ep[j,1] ; j = j + 1 ; endo ; y = x*tb + 2*ep ; Basic-18

19 @ OLS using y and b = invpd(x'x)*(x'y); e = y - x*b ; s2 = (e'e)/(tt-kk); v = s2*invpd(x'x); proc nw(uv,l) ; local j,omeo,n,w,ome ; n = rows(uv) ; omeo = uv'uv/n ; j = 1; do while j <= l ; w = 1 - j/(l+1) ; ome = uv[j+1:n,.]'uv[1:n-j,.]/n ; omeo = omeo + w*(ome+ome') ; j = j + 1 ; endo ; retp(omeo) ; endp ; vc = invpd(x'x)*(tt*nw(x.*e,bandw))*invpd(x'x); se = sqrt(diag(v)) ; sec = sqrt(diag(vc)) ; storb[i,1] = b[2,1]; storse[i,1] = se[2,1]; storsec[i,1] = sec[2,1]; olst = (b[2,1]-tb[2,1])/se[2,1]; df = tt-2; pval = 2*cdftc(abs(olst),df); if pval >= 0.05; stort[i,1]= 0; else; stort[i,1] = 1; endif; olst = (b[2,1]-tb[2,1])/sec[2,1]; df = tt-2; pval = 2*cdftc(abs(olst),df); if pval >= 0.05; stortc[i,1]= 0; else; stortc[i,1] = 1; GLS using y and Basic-19

20 gxx = y[1:rows(y)-1,1]~x[2:rows(x),.]~x[1:rows(x)-1,2]; gyy = y[2:rows(y),1]; durb = invpd(gxx'gxx)*(gxx'gyy); drho = durb[1,1]; gyy = (sqrt(1-drho^2)*y[1,.]) (y[2:rows(y),.]-drho*y[1:rows(y)-1,.]); gxx = (sqrt(1-drho^2)*x[1,.]) (x[2:rows(y),.]-drho*x[1:rows(y)-1,.]); pwglsb = invpd(gxx'gxx)*(gxx'gyy); pwe = gyy - gxx*pwglsb; pws2 = (pwe'pwe)/(tt-kk); pwv = pws2*invpd(gxx'gxx); pwse = sqrt(diag(pwv)) ; storpwgb[i,1] = pwglsb[2,1]; storpwgse[i,1] = pwse[2,1]; glst = (pwglsb[2,1]-tb[2,1])/pwse[2,1]; df = tt-2; pval = 2*cdftc(abs(glst),df); if pval >= 0.05; storpwgt[i,1]= 0; else; storpwgt[i,1] = 1; endif; gyy = gyy[2:rows(gyy),.]; gxx = gxx[2:rows(gxx),.]; coglsb = invpd(gxx'gxx)*(gxx'gyy); coe = gyy - gxx*coglsb; cos2 = (coe'coe)/(tt-kk-1); cov = cos2*invpd(gxx'gxx); cose = sqrt(diag(cov)) ; storcogb[i,1] = coglsb[2,1]; storcogse[i,1] = cose[2,1]; glst = (coglsb[2,1]-tb[2,1])/cose[2,1]; df = tt-2-1; pval = 2*cdftc(abs(glst),df); if pval >= 0.05; storcogt[i,1]= 0; else; storcogt[i,1] = 1; endif; i = i + 1; endo; Basic-20

21 @ Reporting Monte Carlo output file = automonte.out reset; format /rd 12,3; "Monte Carlo results"; "" ; " "; "Checking Unibiasedness"; "-----"; "Mean of OLS b(2) =" meanc(storb); "s.e. of OLS b(2) =" stdc(storb); "mean of estimated s.e. of OLS b(2) =" meanc(storse) ; "mean fo NW-corrected s.e. of OLS b(2) =" meanc(storsec) ; " "; "Mean of P-W feasible GLS b(2) =" meanc(storpwgb); "s.e. of P-W feasible GLS b(2) =" stdc(storpwgb); "mean of P-W estimated s.e. of feasible GLS b(2) =" meanc(storpwgse) ; " "; "Mean of C-O feasible GLS b(2) =" meanc(storcogb); "s.e. of C-O feasible GLS b(2) =" stdc(storcogb); "mean of C-O estimated s.e. of feasible GLS b(2) =" meanc(storcogse) ; ""; "Checking sizes of t-tests at 5% of sig. level"; " "; "Rejection Rate of the t-test with usual OLS =" meanc(stort); "Rejection Rate of the t-test with NW-corrected OLS =" meanc(stortc); "Rejection Rate of the t-test with PW feasible GLS =" meanc(storpwgt); "Rejection Rate of the t-test with Co feasible GLS =" meanc(storcogt); output off ; Basic-21

22 Output file: automonte.out Monte Carlo results Checking Unibiasedness Mean of OLS b(2) = s.e. of OLS b(2) = mean of estimated s.e. of OLS b(2) = mean fo NW-corrected s.e. of OLS b(2) = Mean of P-W feasible GLS b(2) = s.e. of P-W feasible GLS b(2) = mean of P-W estimated s.e. of feasible GLS b(2) = Mean of C-O feasible GLS b(2) = s.e. of C-O feasible GLS b(2) = mean of C-O estimated s.e. of feasible GLS b(2) = Checking sizes of t-tests at 5% of sig. level Rejection Rate of the t-test with usual OLS = Rejection Rate of the t-test with NW-corrected OLS = Rejection Rate of the t-test with PW feasible GLS = Rejection Rate of the t-test with Co feasible GLS = Basic-22