//////////////////////////////////////////////////////////////////////////////// // STATA FOR Hu, Y., Moffitt, R., & Sasaki, Y. (2019): Semiparametric // Estimation of the Canonical PermanentâTransitory Model of Earnings // Dynamics. Quantitative Economics 10 (4), 1495-1536. // // Use it when you consider a state space model where the observed process is an // sum of permanent and transitory components. The command draws the densities // of the perment and transitory components . //////////////////////////////////////////////////////////////////////////////// program define cdecompose, eclass version 14.2 syntax varlist(min=2 numeric) [if] [in] [, p(real 1) q(real 1) delta(real 5) nboot(real 1000)] marksample touse gettoken depvar indepvars : varlist _fv_check_depvar `depvar' fvexpand `indepvars' local cnames `r(varlist)' tempname N b V rho mu kappa mata: estimate_moments("`depvar'", "`cnames'", /// `p', `q', /// `delta', `nboot', /// "`touse'", "`N'", /// "`b'", "`V'", "`rho'", "`mu'", "`kappa'") mata: bootstrap_moments("`depvar'", "`cnames'", /// `p', `q', /// `delta', `nboot', /// "`touse'", "`N'", /// "`b'", "`V'") local cnames U_Mean U_Std_Dev U_Skewness U_Kurtosis V_Mean V_Std_Dev V_Skewness V_Kurtosis matrix colnames `b' = `cnames' matrix colnames `V' = `cnames' matrix rownames `V' = `cnames' ereturn post `b' `V', esample(`touse') buildfvinfo ereturn scalar N = `N' ereturn scalar p = `p' ereturn scalar q = `q' ereturn local cmd "cdecompose" ereturn matrix rho = `rho' ereturn display end mata: //////////////////////////////////////////////////////////////////////////////// // Estimation void estimate_moments( string scalar y1v, string scalar y2v, real scalar p, real scalar q, real scalar delta, real scalar num_boot, string scalar touse, string scalar nname, string scalar bname, string scalar vname, string scalar rhoname, string scalar muname, string scalar kappaname) { printf("\n{hline 78}\n") printf("Executing: Hu, Y., Moffitt, R., & Sasaki, Y. (2019): Semiparametric Estimation\n") printf(" of the Canonical PermanentâTransitory Model of Earnings Dynamics. \n") printf(" Quantitative Economics 10 (4), 1495-1536. \n") printf("{hline 78}\n") y = st_data(., y1v, touse), st_data(., y2v, touse) n = rows(y) // Set main index (t) tdx = p+q+1 //tdx // Check the following is true if( cols(y) < p+q+2+q ){ printf("Error: The number of variables must be at least p+2q+2=%f.\n", p+2*q+2) } // Demean y mean_y = mean(y[.,tdx]) y = y :- mean_y // Set the 100 times the precision of numerical derivative in the frequency domain //delta = 100 // Set slist slist = (-trunc(5*delta)..trunc(5*delta)) / 100 slist_interval = slist[2] - slist[1] printf("Estimating the moments\n") //////////////////////////////////////////////////////////////////////////// // Estimate rho if( p > 0 ){ pprime = 0 Delta0 = y[.,tdx+1-pprime]:-y[.,tdx-q]:-y[.,tdx-pprime]:+y[.,tdx-q-1]:+y[.,tdx-q]:-y[.,tdx-q-1] pprime = 1 Delta = y[.,tdx+1-pprime]:-y[.,tdx-q]:-y[.,tdx-pprime]:+y[.,tdx-q-1]:+y[.,tdx-q]:-y[.,tdx-q-1] if( p > 1 ){ for( pprime = 2 ; pprime <= p ; pprime++ ){ Delta = Delta, y[.,tdx+1-pprime]:-y[.,tdx-q]:-y[.,tdx-pprime]:+y[.,tdx-q-1]:+y[.,tdx-q]:-y[.,tdx-q-1] } } rho = luinv( (y[.,tdx :- ((q+1)..(p+q))])' * Delta :+ 0.05:*diag(J(p,1,n)) ) * (y[.,tdx :- ((q+1)..(p+q))])' * Delta0 // rho } //////////////////////////////////////////////////////////////////////////// // Estimate mu mu = y[.,tdx+q+1] - y[.,tdx] if( p > 0 ){ for( pprime = 1 ; pprime <= p ; pprime++ ){ mu = mu :- rho[pprime] :* ( y[.,tdx+q+1-pprime] - y[.,tdx] ) } } //////////////////////////////////////////////////////////////////////////// // Estimate kappa kappa = -1 if( p > 0 ){ for( pprime = 1 ; pprime <= p ; pprime++ ){ kappa = kappa + rho[pprime] } } //kappa //////////////////////////////////////////////////////////////////////////// // Estimate phi_v phi_v_integrand = slist :* (0+0i) for( idx = 1 ; idx <= length(slist) ; idx++ ){ s = slist[idx] phi_v_integrand[idx] = sum( (0+1i) :* mu :* exp( (0+1i):*s:*y[.,tdx] ) ) / sum( kappa :* exp( (0+1i):*s:*y[.,tdx] ) ) } //phi_v_integrand' phi_v = slist :* (0+0i) middle = trunc(length(slist) / 2) + 1 for( idx = middle + 1 ; idx <= length(slist) ; idx++ ){ phi_v[idx] = phi_v[idx-1] + phi_v_integrand[idx] * slist_interval } for( idx = middle - 1 ; idx >= 1 ; idx-- ){ phi_v[idx] = phi_v[idx+1] - phi_v_integrand[idx] * slist_interval } phi_v = exp( phi_v ) //phi_v' //////////////////////////////////////////////////////////////////////////// // Estimate phi_u phi_y = slist :* (0+0i) for( idx = 1 ; idx <= length(slist) ; idx++ ){ s = slist[idx] phi_y[idx] = sum( exp( (0+1i):*s:*y[.,tdx] ) ) / n } phi_u = phi_y :/ phi_v //phi_u' //////////////////////////////////////////////////////////////////////////// // Estimate derivatives of the characteristic functions phi_u_d1 = slist :* (0+0i) phi_v_d1 = slist :* (0+0i) phi_u_d2 = slist :* (0+0i) phi_v_d2 = slist :* (0+0i) phi_u_d3 = slist :* (0+0i) phi_v_d3 = slist :* (0+0i) phi_u_d4 = slist :* (0+0i) phi_v_d4 = slist :* (0+0i) for( idx = delta+1 ; idx <= length(slist)-delta ; idx++ ){ phi_u_d1[idx] = (phi_u[idx+delta]-phi_u[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d1[idx] = (phi_v[idx+delta]-phi_v[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d1', phi_v_d1' for( idx = 2*delta+1 ; idx <= length(slist)-2*delta ; idx++ ){ phi_u_d2[idx] = (phi_u_d1[idx+delta]-phi_u_d1[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d2[idx] = (phi_v_d1[idx+delta]-phi_v_d1[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d2', phi_v_d2' for( idx = 3*delta+1 ; idx <= length(slist)-3*delta ; idx++ ){ phi_u_d3[idx] = (phi_u_d2[idx+delta]-phi_u_d2[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d3[idx] = (phi_v_d2[idx+delta]-phi_v_d2[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d3', phi_v_d3' for( idx = 4*delta+1 ; idx <= length(slist)-4*delta ; idx++ ){ phi_u_d4[idx] = (phi_u_d3[idx+delta]-phi_u_d3[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d4[idx] = (phi_v_d3[idx+delta]-phi_v_d3[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d4', phi_v_d4' //////////////////////////////////////////////////////////////////////////// // Estimate moments u_m1 = sum( phi_u_d1[((middle-delta)..(middle+delta))] ) / (0+1i) / (2*delta+1) u_m2 = sum( phi_u_d2[((middle-delta)..(middle+delta))] ) / (-1+0i) / (2*delta+1) u_m3 = sum( phi_u_d3[((middle-delta)..(middle+delta))] ) / (0-1i) / (2*delta+1) u_m4 = sum( phi_u_d4[((middle-delta)..(middle+delta))] ) / (1+0i) / (2*delta+1) v_m1 = sum( phi_v_d1[((middle-delta)..(middle+delta))] ) / (0+1i) / (2*delta+1) v_m2 = sum( phi_v_d2[((middle-delta)..(middle+delta))] ) / (-1+0i) / (2*delta+1) v_m3 = sum( phi_v_d3[((middle-delta)..(middle+delta))] ) / (0-1i) / (2*delta+1) v_m4 = sum( phi_v_d4[((middle-delta)..(middle+delta))] ) / (1+0i) / (2*delta+1) //u_m1, u_m2, u_m3, u_m4 //v_m1, v_m2, v_m3, v_m4 mean_u = u_m1 sd_u = ( u_m2 - u_m1^2 )^0.5 skew_u = ( u_m3 - 3*u_m2*mean_u + 3*u_m1*mean_u^2 - mean_u^3 ) / sd_u^3 kurt_u = ( u_m4 - 4*u_m3*mean_u + 6*u_m2*mean_u^2 - 4*u_m1*mean_u^3 + mean_u^4 ) / sd_u^4 mean_v = v_m1 sd_v = ( v_m2 - v_m1^2 )^0.5 skew_v = ( v_m3 - 3*v_m2*mean_v + 3*v_m1*mean_v^2 - mean_v^3 ) / sd_v^3 kurt_v = ( v_m4 - 4*v_m3*mean_v + 6*v_m2*mean_v^2 - 4*v_m1*mean_v^3 + mean_v^4 ) / sd_v^4 //mean_u, sd_u, skew_u, kurt_u //mean_v, sd_v, skew_v, kurt_v b = mean_u, abs(sd_u), skew_u, abs(kurt_u), mean_v, abs(sd_v), skew_v, abs(kurt_v) b = Re(b) //////////////////////////////////////////////////////////////////////////// // Set st_numscalar(nname, n) st_matrix(bname, b) st_matrix(rhoname, rho) st_matrix(muname, mu) st_numscalar(kappaname, kappa) } //////////////////////////////////////////////////////////////////////////////// // Bootstrap void bootstrap_moments( string scalar y1v, string scalar y2v, real scalar p, real scalar q, real scalar delta, real scalar num_boot, string scalar touse, string scalar nname, string scalar bname, string scalar vname) { yy = st_data(., y1v, touse), st_data(., y2v, touse) n = rows(yy) // Set main index (t) tdx = p+q+1 //tdx // Demean y mean_yy = mean(yy[.,tdx]) yy = yy :- mean_yy // Set the 100 times the precision of numerical derivative in the frequency domain //delta = 100 // Set slist slist = (-trunc(5*delta)..trunc(5*delta)) / 100 slist_interval = slist[2] - slist[1] printf("Estimating the variances of the moment estimates\n") //num_boot = 10 blist = J(num_boot,8,0) for( boot_idx = 1 ; boot_idx <= num_boot ; boot_idx++ ){ if( trunc(boot_idx/trunc(num_boot/10)) == boot_idx/trunc(num_boot/10) ){ printf(" %f%%\n",boot_idx/num_boot*100) } indices = trunc(uniform(n,1):*n):+1 y = yy[indices,.] //////////////////////////////////////////////////////////////////////////// // Estimate rho if( p > 0 ){ pprime = 0 Delta0 = y[.,tdx+1-pprime]:-y[.,tdx-q]:-y[.,tdx-pprime]:+y[.,tdx-q-1]:+y[.,tdx-q]:-y[.,tdx-q-1] pprime = 1 Delta = y[.,tdx+1-pprime]:-y[.,tdx-q]:-y[.,tdx-pprime]:+y[.,tdx-q-1]:+y[.,tdx-q]:-y[.,tdx-q-1] if( p > 1 ){ for( pprime = 2 ; pprime <= p ; pprime++ ){ Delta = Delta, y[.,tdx+1-pprime]:-y[.,tdx-q]:-y[.,tdx-pprime]:+y[.,tdx-q-1]:+y[.,tdx-q]:-y[.,tdx-q-1] } } rho = luinv( (y[.,tdx :- ((q+1)..(p+q))])' * Delta :+ 0.05:*diag(J(p,1,n)) ) * (y[.,tdx :- ((q+1)..(p+q))])' * Delta0 // rho } //////////////////////////////////////////////////////////////////////////// // Estimate mu mu = y[.,tdx+q+1] - y[.,tdx] if( p > 0 ){ for( pprime = 1 ; pprime <= p ; pprime++ ){ mu = mu :- rho[pprime] :* ( y[.,tdx+q+1-pprime] - y[.,tdx] ) } } //////////////////////////////////////////////////////////////////////////// // Estimate kappa kappa = -1 if( p > 0 ){ for( pprime = 1 ; pprime <= p ; pprime++ ){ kappa = kappa + rho[pprime] } } //kappa //////////////////////////////////////////////////////////////////////////// // Estimate phi_v phi_v_integrand = slist :* (0+0i) for( idx = 1 ; idx <= length(slist) ; idx++ ){ s = slist[idx] phi_v_integrand[idx] = sum( (0+1i) :* mu :* exp( (0+1i):*s:*y[.,tdx] ) ) / sum( kappa :* exp( (0+1i):*s:*y[.,tdx] ) ) } //phi_v_integrand' phi_v = slist :* (0+0i) middle = trunc(length(slist) / 2) + 1 for( idx = middle + 1 ; idx <= length(slist) ; idx++ ){ phi_v[idx] = phi_v[idx-1] + phi_v_integrand[idx] * slist_interval } for( idx = middle - 1 ; idx >= 1 ; idx-- ){ phi_v[idx] = phi_v[idx+1] - phi_v_integrand[idx] * slist_interval } phi_v = exp( phi_v ) //phi_v' //////////////////////////////////////////////////////////////////////////// // Estimate phi_u phi_y = slist :* (0+0i) for( idx = 1 ; idx <= length(slist) ; idx++ ){ s = slist[idx] phi_y[idx] = sum( exp( (0+1i):*s:*y[.,tdx] ) ) / n } phi_u = phi_y :/ phi_v //phi_u' //////////////////////////////////////////////////////////////////////////// // Estimate derivatives of the characteristic functions phi_u_d1 = slist :* (0+0i) phi_v_d1 = slist :* (0+0i) phi_u_d2 = slist :* (0+0i) phi_v_d2 = slist :* (0+0i) phi_u_d3 = slist :* (0+0i) phi_v_d3 = slist :* (0+0i) phi_u_d4 = slist :* (0+0i) phi_v_d4 = slist :* (0+0i) for( idx = delta+1 ; idx <= length(slist)-delta ; idx++ ){ phi_u_d1[idx] = (phi_u[idx+delta]-phi_u[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d1[idx] = (phi_v[idx+delta]-phi_v[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d1', phi_v_d1' for( idx = 2*delta+1 ; idx <= length(slist)-2*delta ; idx++ ){ phi_u_d2[idx] = (phi_u_d1[idx+delta]-phi_u_d1[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d2[idx] = (phi_v_d1[idx+delta]-phi_v_d1[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d2', phi_v_d2' for( idx = 3*delta+1 ; idx <= length(slist)-3*delta ; idx++ ){ phi_u_d3[idx] = (phi_u_d2[idx+delta]-phi_u_d2[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d3[idx] = (phi_v_d2[idx+delta]-phi_v_d2[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d3', phi_v_d3' for( idx = 4*delta+1 ; idx <= length(slist)-4*delta ; idx++ ){ phi_u_d4[idx] = (phi_u_d3[idx+delta]-phi_u_d3[idx-delta])/(slist[idx+delta]-slist[idx-delta]) phi_v_d4[idx] = (phi_v_d3[idx+delta]-phi_v_d3[idx-delta])/(slist[idx+delta]-slist[idx-delta]) } //phi_u_d4', phi_v_d4' //////////////////////////////////////////////////////////////////////////// // Estimate moments u_m1 = sum( phi_u_d1[((middle-delta)..(middle+delta))] ) / (0+1i) / (2*delta+1) u_m2 = sum( phi_u_d2[((middle-delta)..(middle+delta))] ) / (-1+0i) / (2*delta+1) u_m3 = sum( phi_u_d3[((middle-delta)..(middle+delta))] ) / (0-1i) / (2*delta+1) u_m4 = sum( phi_u_d4[((middle-delta)..(middle+delta))] ) / (1+0i) / (2*delta+1) v_m1 = sum( phi_v_d1[((middle-delta)..(middle+delta))] ) / (0+1i) / (2*delta+1) v_m2 = sum( phi_v_d2[((middle-delta)..(middle+delta))] ) / (-1+0i) / (2*delta+1) v_m3 = sum( phi_v_d3[((middle-delta)..(middle+delta))] ) / (0-1i) / (2*delta+1) v_m4 = sum( phi_v_d4[((middle-delta)..(middle+delta))] ) / (1+0i) / (2*delta+1) //u_m1, u_m2, u_m3, u_m4 //v_m1, v_m2, v_m3, v_m4 mean_u = u_m1 sd_u = ( u_m2 - u_m1^2 )^0.5 skew_u = ( u_m3 - 3*u_m2*mean_u + 3*u_m1*mean_u^2 - mean_u^3 ) / sd_u^3 kurt_u = ( u_m4 - 4*u_m3*mean_u + 6*u_m2*mean_u^2 - 4*u_m1*mean_u^3 + mean_u^4 ) / sd_u^4 mean_v = v_m1 sd_v = ( v_m2 - v_m1^2 )^0.5 skew_v = ( v_m3 - 3*v_m2*mean_v + 3*v_m1*mean_v^2 - mean_v^3 ) / sd_v^3 kurt_v = ( v_m4 - 4*v_m3*mean_v + 6*v_m2*mean_v^2 - 4*v_m1*mean_v^3 + mean_v^4 ) / sd_v^4 //mean_u, sd_u, skew_u, kurt_u //mean_v, sd_v, skew_v, kurt_v b = mean_u, abs(sd_u), skew_u, abs(kurt_u), mean_v, abs(sd_v), skew_v, abs(kurt_v) b = Re(b) blist[boot_idx,.] = b } //sort(colshape(abs(blist),1),1) for( idx = 1 ; idx <= 8 ; idx++ ){ large = blist[.,idx] :>= sort(blist[.,idx],1)[trunc(num_boot*0.995)+1] small = blist[.,idx] :<= sort(blist[.,idx],1)[trunc(num_boot*0.005)+1] blist[.,idx] = (!small :& !large) :* blist[.,idx] :+ small :* sort(blist[.,idx],1)[trunc(num_boot*0.005)+1] + large :* sort(blist[.,idx],1)[trunc(num_boot*0.995)+1] } V = ( ( blist' * blist ) :/ num_boot :- ( J(1,num_boot,1) * blist :/ num_boot)' * ( J(1,num_boot,1) * blist :/ num_boot) ) //////////////////////////////////////////////////////////////////////////// // Set st_numscalar(nname, n) st_matrix(vname, V) printf("\n{hline 78}\n") printf("ARMA(%f,%f) process - p=%f & q=%f \n",p,q,p,q) printf("Number of cross-sectional observations: N=%f\n",n) printf("Number of time periods of input variables: T=%f\n",cols(yy)) printf("Number of minimum time periods required for identification: p+2q+2=%f\n",p+2*q+2) printf("{hline 78}\n") printf("Displayed are moments in the time period corresponding to the %f-th input var.\n",tdx) printf("U = Permanent Component\n") printf("V = Transitory Component\n") } end ////////////////////////////////////////////////////////////////////////////////