Plot of Epsilon over Time -- Case 1 1 Time series data: Part Epsilon - 1 - - - -1 1 51 7 11 1 151 17 Time period Plot of Epsilon over Time -- Case Plot of Epsilon over Time -- Case 3 1 3 1 Epsilon - Epsilon -1 - - - - -3-1 1 51 7 11 1 151 17-1 51 7 11 1 151 17 Time period Time period 3 1
Scatter Plot: Case 1 Scatter Plot: Case Epsilon(t) 1 - - - - -1-1 - - - - 1 Epsilon(t-1) 5 Epsilon(t) 1 - - - - -1-1 - - - - 1 Epsilon(t-1) Epsilon(t) Scatter Plot: Case 3 3 1-1 - -3 - - -3 - -1 1 3 Epsilon(t-1) 7 For <ρ<1 H o : ρ= H a : ρ> ˆ DW Statistic d < d reject null l d dˆ < d uncertain l u dˆ d cannot reject null u
9 1 Phillips Curve, 19-197 Phillips Curve, 1971-7 1 1 1 1 1 Inflation Inflation - 3 5 7 3 5 7 9 1 11 Unemployment rate Unemployment rate 11 1 3
Phillips Curve, 19-7 1 Inflation 1 1 1 * The order of the time series * data must be specified by an * index Here, we use year as the * index tsset year time variable: year, 197 to 7 delta: 1 unit - 3 5 7 9 1 11 Unemployment rate 13 1 * run classical phillips curve reg inflation unemp Source SS df MS Number of obs = -------------+------------------------------ F( 1, 5) = Model 3353573 1 3353573 Prob > F = Residual 7973 5 915 R-squared = 5 -------------+------------------------------ Adj R-squared = 93 Total 513 59 9559 Root MSE = 75 inflation Coef Std Err t P> t [95% Conf Interval] unemp 5131 553 1 9 3371 139 _cons 951991 171 5 51-19979 399 * get durbin watson statistic estat dwatson Durbin-Watson d-statistic(, ) = 75 15 1
* output residual predict resid_a, residual (1 missing value generated) * lag the variable 1 period gen resid_a1=resid_a[_n-1] ( missing values generated) * estimate ar(1) term reg resid_a resid_a1, noconst Source SS df MS Number of obs = 59 -------------+------------------------------ F( 1, 5) = 313 Model 157395 1 157395 Prob > F = Residual 97999 5 513377 R-squared = 319 -------------+------------------------------ Adj R-squared = 335 Total 5915 59 77399 Root MSE = resid_a Coef Std Err t P> t [95% Conf Interval] resid_a1 5 13555 59 3995 77559 17 1 *now estimate augmented Phillips Curve * construct difference in inflation gen inflation_1=inflation[_n-1] (1 missing value generated) gen d_inf=inflation-inflation_1 (1 missing value generated) reg d_inf unemp Lag inflation 1 st Difference in inflation Identified time series ar(k) correction procedure Corcran-Orcutt procedure Source SS df MS Number of obs = -------------+------------------------------ F( 1, 5) = 37 Model 159 1 159 Prob > F = 573 Residual 3331 5 57519 R-squared = 9 -------------+------------------------------ Adj R-squared = 7 Total 355557 59 1331 Root MSE = 393 d_inf Coef Std Err t P> t [95% Conf Interval] unemp -1 1733-19 57-351 131517 _cons 1159 193 17 91-377 57791 estat dwatson prais d_inf unemp, corc twostep Same model statement As before Durbin-Watson d-statistic(, ) = 15911 19 5
prais d_inf unemp, corc twostep Iteration : rho = Iteration 1: rho = 1559 Cochrane-Orcutt AR(1) regression -- twostep estimates Augumented Phillips Curve, 19-7 Source SS df MS Number of obs = 59 -------------+------------------------------ F( 1, 57) = 713 Model 3331 1 3331 Prob > F = 99 Residual 713 57 1971 R-squared = 1111 -------------+------------------------------ Adj R-squared = 955 Total 3919 5 537975 Root MSE = 195 d_inf Coef Std Err t P> t [95% Conf Interval] unemp -5935 3151-7 1-133 -1 _cons 395 193 5 1 7159 55 rho 15599 Durbin-Watson statistic (original) 15911 Durbin-Watson statistic (transformed) 19715 Adjusting for ar(1) changes Inflation - Lagged Inflation - - - - -1-1 3 5 7 9 1 11 Unemployment rate 1 Results some Taylor Rule i r a a y yˆ * t = πt + t + n( πt πt ) + y( t t) i = r a π + (1 + a ) π + a ( y yˆ ) * t t n t n t y t t let π = percent * t r = percent t Taylor suggests a = a = 5 i = 1+ 15π + 5( y yˆ ) t t t t n y 3
* index the data for time series use tsset index *generate ln(gdp) gen gdprl=ln(gdp_r) label var gdprl "ln of real dgp" *get lag of fed fund rate * will use later gen ffr1=ffr[_n-1] label var ffr1 "lag of fed fund rate" *get lag of gdprl * will use later gen gdprl1=gdprl[_n-1] label var gdprl1 "lag of gdprl" * generate 1 year inflation * by taking difference between current * and a period lag * report in percent gen gdp_def=gdp_def[_n-] gen inflation=1*(ln(gdp_def)- ln(gdp_def)) label var inflation "one-year inflation rate in percent" * reduce the data to the post-greenspan years keep if year>=197 5 * run a regression of gdprl on a trend reg gdprl index Source SS df MS Number of obs = -------------+------------------------------ F( 1, ) =17 Model 917537 1 917537 Prob > F = Residual 379 999 R-squared = 993 -------------+------------------------------ Adj R-squared = 99 Total 93551331 5 353551 Root MSE = 1 gdprl Coef Std Err t P> t [95% Conf Interval] index 7137 713 139 7719 75555 _cons 53915 15 113 3771 7113 * output residuals (output gap) predict gdprl_res, residuals * scale gap in percent by * multiplying by 1 gen gap=1*gdprl_res * run taylor rule regression reg ffr inflation gap Source SS df MS Number of obs = -------------+------------------------------ F(, 3) = 197 Model 153791 39 Prob > F = Residual 971 3 31513 R-squared = 37 -------------+------------------------------ Adj R-squared = 3 Total 3319 5 511 Root MSE = 17 ffr Coef Std Err t P> t [95% Conf Interval] inflation 113 535 13 1 gap 5193 1135 35 7953 7539 _cons 1 53575 315 7171 33159 estat dwatson Durbin-Watson d-statistic( 3, ) = 93391 7 7
estat dwatson Durbin-Watson d-statistic( 3, ) = 93391 * test the taylor rule parameters using * an f-test test (gap=5) (inflation=15)(_cons=1) ( 1) gap = 5 ( ) inflation = 15 ( 3) _cons = 1 F( 3, 3) = 9 Prob > F = 9 3 prais ffr inflation gap, corc twostep Iteration : rho = Iteration 1: rho = 93 Cochrane-Orcutt AR(1) regression -- twostep estimates Source SS df MS Number of obs = 5 -------------+------------------------------ F(, ) = 113 Model 7355 37 Prob > F = Residual 133173 1991 R-squared = 15 -------------+------------------------------ Adj R-squared = 1959 Total 53 7195 Root MSE = ffr Coef Std Err t P> t [95% Conf Interval] inflation 755971 93 339 1 31313 119953 gap 311593 9393 33 1 15571 91 _cons 177 1573 13 199-17 775 rho 9 Durbin-Watson statistic (original) 9339 Durbin-Watson statistic (transformed) 951 31