Time series data: Part 2

Transcription

1 Plot of Epsilon over Time -- Case 1 1 Time series data: Part Epsilon Time period Plot of Epsilon over Time -- Case Plot of Epsilon over Time -- Case Epsilon - Epsilon Time period Time period 3 1

2 Scatter Plot: Case 1 Scatter Plot: Case Epsilon(t) Epsilon(t-1) 5 Epsilon(t) Epsilon(t-1) Epsilon(t) Scatter Plot: Case 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

3 9 1 Phillips Curve, Phillips Curve, Inflation Inflation Unemployment rate Unemployment rate

4 Phillips Curve, Inflation * 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 Unemployment rate 13 1 * run classical phillips curve reg inflation unemp Source SS df MS Number of obs = F( 1, 5) = Model Prob > F = Residual R-squared = Adj R-squared = 93 Total Root MSE = 75 inflation Coef Std Err t P> t [95% Conf Interval] unemp _cons * get durbin watson statistic estat dwatson Durbin-Watson d-statistic(, ) =

5 * 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 = F( 1, 5) = 313 Model Prob > F = Residual R-squared = Adj R-squared = 335 Total Root MSE = resid_a Coef Std Err t P> t [95% Conf Interval] resid_a *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 Prob > F = 573 Residual R-squared = Adj R-squared = 7 Total Root MSE = 393 d_inf Coef Std Err t P> t [95% Conf Interval] unemp _cons estat dwatson prais d_inf unemp, corc twostep Same model statement As before Durbin-Watson d-statistic(, ) =

6 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 = F( 1, 57) = 713 Model Prob > F = 99 Residual R-squared = Adj R-squared = 955 Total Root MSE = 195 d_inf Coef Std Err t P> t [95% Conf Interval] unemp _cons rho Durbin-Watson statistic (original) Durbin-Watson statistic (transformed) Adjusting for ar(1) changes Inflation - Lagged Inflation 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

7 * 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 Prob > F = Residual R-squared = Adj R-squared = 99 Total Root MSE = 1 gdprl Coef Std Err t P> t [95% Conf Interval] index _cons * 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 Prob > F = Residual R-squared = Adj R-squared = 3 Total Root MSE = 17 ffr Coef Std Err t P> t [95% Conf Interval] inflation gap _cons estat dwatson Durbin-Watson d-statistic( 3, ) =

8 estat dwatson Durbin-Watson d-statistic( 3, ) = * 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 = F(, ) = 113 Model Prob > F = Residual R-squared = Adj R-squared = 1959 Total Root MSE = ffr Coef Std Err t P> t [95% Conf Interval] inflation gap _cons rho 9 Durbin-Watson statistic (original) 9339 Durbin-Watson statistic (transformed)