Econometrics Lab Hour Session 6
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1 Econometrics Lab Hour Session 6 Agustín Bénétrix benetria@tcd.ie Office hour: Wednesday 4-5 Room 3021 Martin Schmitz schmitzm@tcd.ie Office hour: Monday 5-6 Room 3021
2 Outline Importing the dataset Time Series Unit roots Spurious regressions Cointegration Project time
3 We have to Confirm that lsp500 and lip contain unit roots using Augmented Dickey-Fuller test Need logs and log changes 4 lags With and without time trend H 0 : Series contains unit root
4 We need to generate and run LSP500 = log(sp500) DLSP500 = LSP500 - LSP500(-1) Run: DLSP500 CONSTANT LSP500(-1) DLSP500(-1) DLSP500(-2) DLSP500(-3) DLSP500(-4) Run: DLSP500 CONSTANT TREND LSP500(-1) DLSP500(-1) DLSP500(-2) DLSP500(-3) DLSP500(-4)
5 We need to generate and run LIP = log(ip) DLIP = LIP - LIP (-1) Run: DLIP CONSTANT LIP(-1) DLIP(-1) DLIP(-2) DLIP(-3) DLIP(-4) Run: DLIP CONSTANT TREND LIP(-1) DLIP(-1) DLIP(-2) DLIP(-3) DLIP(-4)
6 Output (LSP500) Ordinary Least Squares Estimation Dependent variable is DLSP observations used for estimation from 1947M7 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT [.154] LSP500(-1) [.428] DLSP500(-1) [.000] DLSP500(-2) [.060] DLSP500(-3) [.701] DLSP500(-4) [.303] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 5, 547) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
7 Output (with time trend) (LSP500) Ordinary Least Squares Estimation Dependent variable is DLSP observations used for estimation from 1947M7 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT [.014] T.6331E E [.040] LSP500(-1) [.028] DLSP500(-1) [.000] DLSP500(-2) [.081] DLSP500(-3) [.616] DLSP500(-4) [.232] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 6, 546) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
8 Output (LIP) Ordinary Least Squares Estimation Dependent variable is DLIP 553 observations used for estimation from 1947M7 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT [.079] LIP(-1) E [.171] DLIP(-1) [.000] DLIP(-2) [.117] DLIP(-3) [.154] DLIP(-4) [.951] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 5, 547) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
9 Output (with time trend) (LIP) Ordinary Least Squares Estimation Dependent variable is DLIP 553 observations used for estimation from 1947M7 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT [.008] T.3565E E [.021] LIP(-1) [.012] DLIP(-1) [.000] DLIP(-2) [.088] DLIP(-3) [.110] DLIP(-4) [.721] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 6, 546) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
10 Both variables are I(1) For lsp500, the ADF statistic without a trend is t =.79; with a trend, the t statistic is This are both well above their respective 10% critical values. In addition, the estimated roots are quite close to one. For lip, the ADF statistic without a trend is 1.37 without a trend and 2.52 with a trend. Again, these are not close to rejecting even at the 10% levels, and the estimated roots are very close to one. critical value at 10%:-2.57 and (with time trend)
11 spurious regression? Ordinary Least Squares Estimation Dependent variable is LSP observations used for estimation from 1947M7 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT [.000] LIP [.000] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 1, 551) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
12 The t statistic for lip is over 70, and the R-squared is around.90. These are hallmarks of spurious regressions. Spurious regressions includes two independent I(1) time series
13 Test for co-integration Use the residuals to test whether lsp500 and lip are cointegrated Only if both variables are I(1), i.e. have a unit root Dickey Fuller and ADF 2 lags
14 Test for co-integration Save residuals Generate difference ADF test of residuals H 0 : No Co-integration
15 Run again and keep residuals Ordinary Least Squares Estimation Dependent variable is LSP observations used for estimation from 1947M7 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT [.000] LIP [.000] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 1, 551) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
16 Apply ADF to residuals Ordinary Least Squares Estimation Dependent variable is DRESID 550 observations used for estimation from 1947M10 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] RESID(-1) [.119] DRESID(-1) [.000] DRESID(-2) [.165] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 2, 547) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
17 Errors are I(1), -> no CO-I The ADF statistic (with two lagged changes) is There is no evidence of cointegration. (The 10% critical value is 3.04.) Add time trend to initial model
18 Add the time trend estimate and keep residuals Ordinary Least Squares Estimation Dependent variable is LSP observations used for estimation from 1947M2 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT [.000] T E [.000] LIP [.004] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 2, 555) [.000] Mean of Dependent Variable S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
19 Apply ADF to residuals Ordinary Least Squares Estimation Dependent variable is DRESID1 555 observations used for estimation from 1947M5 to 1993M7 Regressor Coefficient Standard Error T-Ratio[Prob] RESID1(-1) [.061] DRESID1(-1) [.000] DRESID1(-2) [.059] R-Squared R-Bar-Squared S.E. of Regression F-stat. F( 2, 552) [.000] Mean of Dependent Variable.9645E-3 S.D. of Dependent Variable Residual Sum of Squares Equation Log-likelihood Akaike Info. Criterion Schwarz Bayesian Criterion DW-statistic
20 After adding a linear time trend to the regression, the ADF statistic applied to the residuals is Even with a time trend there is no evidence of cointegration (critical value at 10% is -3.50) It appears that lsp500 and lip are not cointegrated, even if we allow them to have unrestricted linear time trends. The analysis does not point to a long-run equilibrium relationship.
21 Project time.
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