# Econometrics

Econometrics Essay Help

• Generate two new variables, log of stock price, log(Pt), and log of production, log(Yt). Draw line plots for the time series variables, log(Pt) and log(Yt) separately.

Table 1:

• Perform Augmented Dickey‐Fuller (ADF) test for log(Pt):
1. with three lagged changes and intercept

Table 3: Augmented Dickey‐Fuller (ADF) test for with three lagged changes and intercept

 Null Hypothesis: LOG_PT has a unit root Exogenous: Constant Lag Length: 1 (Automatic – based on SIC, maxlag=3) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -0.532353 0.8820 Test critical values: 1% level -3.441882 5% level -2.866519 10% level -2.569482 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOG_PT) Method: Least Squares Date: 05/08/15   Time: 09:18 Sample (adjusted): 1967M03 2013M06 Included observations: 556 after adjustments Variable Coefficient Std. Error t-Statistic Prob. LOG_PT(-1) -0.000854 0.001605 -0.532353 0.5947 D(LOG_PT(-1)) 0.244440 0.041199 5.933137 0.0000 C 0.008307 0.007227 1.149437 0.2509 R-squared 0.060136 Mean dependent var 0.006016 Adjusted R-squared 0.056737 S.D. dependent var 0.033716 S.E. of regression 0.032745 Akaike info criterion -3.994730 Sum squared resid 0.592959 Schwarz criterion -3.971417 Log likelihood 1113.535 Hannan-Quinn criter. -3.985624 F-statistic 17.69148 Durbin-Watson stat 1.949516 Prob(F-statistic) 0.000000

1. b) with three lagged changes, intercept and trend and interpret the result.

Table 4: Augmented Dickey‐Fuller (ADF) test for with three lagged changes, intercept and trend

 Null Hypothesis: LOG_PT has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic – based on SIC, maxlag=3) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.054036 0.5698 Test critical values: 1% level -3.974647 5% level -3.417923 10% level -3.131415 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOG_PT) Method: Least Squares Date: 05/08/15   Time: 09:23 Sample (adjusted): 1967M03 2013M06 Included observations: 556 after adjustments Variable Coefficient Std. Error t-Statistic Prob. LOG_PT(-1) -0.011618 0.005656 -2.054036 0.0404 D(LOG_PT(-1)) 0.249793 0.041179 6.066076 0.0000 C 0.038938 0.017038 2.285356 0.0227 @TREND(1967M01) 6.05E-05 3.05E-05 1.984088 0.0477 R-squared 0.066791 Mean dependent var 0.006016 Adjusted R-squared 0.061719 S.D. dependent var 0.033716 S.E. of regression 0.032659 Akaike info criterion -3.998240 Sum squared resid 0.588760 Schwarz criterion -3.967155 Log likelihood 1115.511 Hannan-Quinn criter. -3.986098 F-statistic 13.16916 Durbin-Watson stat 1.951767 Prob(F-statistic) 0.000000
• Repeat (ii) for log(Yt)

Table 5: Augmented Dickey‐Fuller (ADF) test for with three lagged changes and intercept

 Null Hypothesis: LOG_YT has a unit root Exogenous: Constant Lag Length: 1 (Automatic – based on SIC, maxlag=3) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -1.417834 0.5744 Test critical values: 1% level -3.441882 5% level -2.866519 10% level -2.569482 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOG_YT) Method: Least Squares Date: 05/08/15   Time: 09:26 Sample (adjusted): 1967M03 2013M06 Included observations: 556 after adjustments Variable Coefficient Std. Error t-Statistic Prob. LOG_YT(-1) -0.001250 0.000881 -1.417834 0.1568 D(LOG_YT(-1)) 0.391739 0.039063 10.02838 0.0000 C 0.006766 0.003583 1.888183 0.0595 R-squared 0.158781 Mean dependent var 0.002856 Adjusted R-squared 0.155739 S.D. dependent var 0.010989 S.E. of regression 0.010097 Akaike info criterion -6.347810 Sum squared resid 0.056376 Schwarz criterion -6.324497 Log likelihood 1767.691 Hannan-Quinn criter. -6.338704 F-statistic 52.18973 Durbin-Watson stat 2.074044 Prob(F-statistic) 0.000000

Table 6: Augmented Dickey‐Fuller (ADF) test for with three lagged changes, intercept and trend

 Null Hypothesis: LOG_YT has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic – based on SIC, maxlag=3) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.015104 0.5913 Test critical values: 1% level -3.974647 5% level -3.417923 10% level -3.131415 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOG_YT) Method: Least Squares Date: 05/08/15   Time: 09:27 Sample (adjusted): 1967M03 2013M06 Included observations: 556 after adjustments Variable Coefficient Std. Error t-Statistic Prob. LOG_YT(-1) -0.010037 0.004981 -2.015104 0.0444 D(LOG_YT(-1)) 0.398009 0.039142 10.16841 0.0000 C 0.034568 0.015918 2.171599 0.0303 @TREND(1967M01) 2.71E-05 1.51E-05 1.792372 0.0736 R-squared 0.163649 Mean dependent var 0.002856 Adjusted R-squared 0.159103 S.D. dependent var 0.010989 S.E. of regression 0.010077 Akaike info criterion -6.350016 Sum squared resid 0.056050 Schwarz criterion -6.318931 Log likelihood 1769.304 Hannan-Quinn criter. -6.337875 F-statistic 36.00323 Durbin-Watson stat 2.082319 Prob(F-statistic) 0.000000
• Run the following simple regression,

and discuss the result in relation with (ii) and (iii)

Table 7: Simple Regression

 Dependent Variable: LOG_PT Method: Least Squares Date: 05/08/15   Time: 09:55 Sample: 1967M01 2013M06 Included observations: 558 LOG_PT=C(1)+C(2)*LOG_YT Coefficient Std. Error t-Statistic Prob. C(1) -2.401727 0.095486 -25.15256 0.0000 C(2) 1.694074 0.023540 71.96560 0.0000 R-squared 0.903052 Mean dependent var 4.420099 Adjusted R-squared 0.902878 S.D. dependent var 0.870710 S.E. of regression 0.271352 Akaike info criterion 0.232777 Sum squared resid 40.93929 Schwarz criterion 0.248276 Log likelihood -62.94473 Hannan-Quinn criter. 0.238830 F-statistic 5179.048 Durbin-Watson stat 0.019658 Prob(F-statistic) 0.000000

• Use the residuals from the regression in (iv) to test whether log(Pt) and log(Yt) are cointegrated. Use the ADF with two lags and intercept. What do you conclude?

Table 8: Augmented Dickey‐Fuller (ADF) test for residuals with two lagged changes and intercept

 Null Hypothesis: RESID01 has a unit root Exogenous: Constant Lag Length: 1 (Automatic – based on SIC, maxlag=2) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -1.711141 0.4251 Test critical values: 1% level -3.441882 5% level -2.866519 10% level -2.569482 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID01) Method: Least Squares Date: 05/08/15   Time: 10:08 Sample (adjusted): 1967M03 2013M06 Included observations: 556 after adjustments Variable Coefficient Std. Error t-Statistic Prob. RESID01(-1) -0.009901 0.005786 -1.711141 0.0876 D(RESID01(-1)) 0.264542 0.041135 6.431000 0.0000 C 0.000849 0.001558 0.545010 0.5860 R-squared 0.071524 Mean dependent var 0.001178 Adjusted R-squared 0.068166 S.D. dependent var 0.038039 S.E. of regression 0.036720 Akaike info criterion -3.765623 Sum squared resid 0.745632 Schwarz criterion -3.742310 Log likelihood 1049.843 Hannan-Quinn criter. -3.756517 F-statistic 21.29997 Durbin-Watson stat 1.957045 Prob(F-statistic) 0.000000
• Run the following simple regression with a linear time trend t,

and test for cointegration using the same tests from (v). What do you conclude?

Table 9: Simple regression with a linear time trend t

 Dependent Variable: LOG_PT Method: Least Squares Date: 05/08/15   Time: 10:28 Sample: 1967M01 2013M06 Included observations: 558 LOG_PT=C(1)+C(2)*LOG_YT+C(3)*T Coefficient Std. Error t-Statistic Prob. C(1) 1.873766 0.383551 4.885311 0.0000 C(2) 0.343890 0.119929 2.867437 0.0043 C(3) 0.004156 0.000363 11.43819 0.0000 R-squared 0.921546 Mean dependent var 4.420099 Adjusted R-squared 0.921264 S.D. dependent var 0.870710 S.E. of regression 0.244321 Akaike info criterion 0.024696 Sum squared resid 33.12954 Schwarz criterion 0.047945 Log likelihood -3.890231 Hannan-Quinn criter. 0.033776 F-statistic 3259.623 Durbin-Watson stat 0.019204 Prob(F-statistic) 0.000000

Table 10: Augmented Dickey‐Fuller (ADF) test for residuals with two lagged changes and intercept

 Null Hypothesis: RESID02 has a unit root Exogenous: Constant Lag Length: 1 (Automatic – based on SIC, maxlag=2) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -1.987547 0.2924 Test critical values: 1% level -3.441882 5% level -2.866519 10% level -2.569482 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID02) Method: Least Squares Date: 05/08/15   Time: 10:38 Sample (adjusted): 1967M03 2013M06 Included observations: 556 after adjustments Variable Coefficient Std. Error t-Statistic Prob. RESID02(-1) -0.011370 0.005720 -1.987547 0.0474 D(RESID02(-1)) 0.246248 0.041209 5.975565 0.0000 C 0.000647 0.001390 0.465665 0.6416 R-squared 0.064490 Mean dependent var 0.000878 Adjusted R-squared 0.061107 S.D. dependent var 0.033818 S.E. of regression 0.032769 Akaike info criterion -3.993316 Sum squared resid 0.593798 Schwarz criterion -3.970003 Log likelihood 1113.142 Hannan-Quinn criter. -3.984210 F-statistic 19.06075 Durbin-Watson stat 1.949210 Prob(F-statistic) 0.000000

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Q2

• Draw line plots for the price (Pt) and the first difference of the price (ΔPt) over time. Perform the ADF test for the price (Pt) and the first difference of the price (ΔPt) with intercept. (Choose the automatic selection option for the lag length of the augmented term based on the Schwarz Information Criterion.) Interpret the results.

Table 11: Line Plot for P

Table 12: Line Plot for first difference of P

Table 13: ADF test for the price (Pt) with intercept

 Null Hypothesis: P has a unit root Exogenous: Constant Lag Length: 8 (Automatic – based on SIC, maxlag=13) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -1.663798 0.4478 Test critical values: 1% level -3.468749 5% level -2.878311 10% level -2.575791 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(P) Method: Least Squares Date: 05/08/15   Time: 10:55 Sample (adjusted): 10 180 Included observations: 171 after adjustments Variable Coefficient Std. Error t-Statistic Prob. P(-1) -0.018336 0.011021 -1.663798 0.0981 D(P(-1)) 0.493112 0.073918 6.671073 0.0000 D(P(-2)) 0.132977 0.083177 1.598726 0.1118 D(P(-3)) 0.186893 0.083911 2.227271 0.0273 D(P(-4)) -0.489934 0.083849 -5.843036 0.0000 D(P(-5)) 0.217306 0.083244 2.610463 0.0099 D(P(-6)) 0.035399 0.083934 0.421747 0.6738 D(P(-7)) 0.132032 0.083994 1.571927 0.1179 D(P(-8)) -0.305237 0.075267 -4.055388 0.0001 C 0.000779 0.000481 1.619769 0.1072 R-squared 0.445381 Mean dependent var 0.000124 Adjusted R-squared 0.414378 S.D. dependent var 0.003970 S.E. of regression 0.003038 Akaike info criterion -8.698325 Sum squared resid 0.001486 Schwarz criterion -8.514602 Log likelihood 753.7068 Hannan-Quinn criter. -8.623778 F-statistic 14.36550 Durbin-Watson stat 1.930472 Prob(F-statistic) 0.000000

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Table 14: ADF test for the first difference of the price (ΔPt) with intercept

 Null Hypothesis: P_1ST_DIFF has a unit root Exogenous: Constant Lag Length: 7 (Automatic – based on SIC, maxlag=13) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -6.108588 0.0000 Test critical values: 1% level -3.468749 5% level -2.878311 10% level -2.575791 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(P_1ST_DIFF) Method: Least Squares Date: 05/08/15   Time: 10:58 Sample (adjusted): 9 179 Included observations: 171 after adjustments Variable Coefficient Std. Error t-Statistic Prob. P_1ST_DIFF(-1) -0.664723 0.108818 -6.108588 0.0000 D(P_1ST_DIFF(-1)) 0.159649 0.108726 1.468364 0.1439 D(P_1ST_DIFF(-2)) 0.284510 0.103138 2.758548 0.0065 D(P_1ST_DIFF(-3)) 0.460876 0.097998 4.702924 0.0000 D(P_1ST_DIFF(-4)) -0.045385 0.088750 -0.511381 0.6098 D(P_1ST_DIFF(-5)) 0.171096 0.087304 1.959773 0.0517 D(P_1ST_DIFF(-6)) 0.200063 0.083150 2.406038 0.0173 D(P_1ST_DIFF(-7)) 0.322949 0.074916 4.310818 0.0000 C 7.86E-05 0.000234 0.336045 0.7373 R-squared 0.417913 Mean dependent var 2.22E-05 Adjusted R-squared 0.389168 S.D. dependent var 0.003909 S.E. of regression 0.003055 Akaike info criterion -8.692973 Sum squared resid 0.001512 Schwarz criterion -8.527622 Log likelihood 752.2492 Hannan-Quinn criter. -8.625881 F-statistic 14.53863 Durbin-Watson stat 1.934112 Prob(F-statistic) 0.000000

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Table 15: Simple linear regression

 Dependent Variable: P Method: Least Squares Date: 05/08/15   Time: 11:27 Sample: 1 180 Included observations: 180 P= C(1) + C(2)*M Coefficient Std. Error t-Statistic Prob. C(1) 0.019366 0.004147 4.670458 0.0000 C(2) 0.232784 0.049548 4.698206 0.0000 R-squared 0.110325 Mean dependent var 0.037300 Adjusted R-squared 0.105327 S.D. dependent var 0.022971 S.E. of regression 0.021728 Akaike info criterion -4.809422 Sum squared resid 0.084031 Schwarz criterion -4.773945 Log likelihood 434.8480 Hannan-Quinn criter. -4.795037 F-statistic 22.07314 Durbin-Watson stat 0.048562 Prob(F-statistic) 0.000005

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• Repeat (i) for the money supply (Mt) and the first difference of the money supply (ΔMt)

Table 16: Line plot for M
Table 17: Line Plot for First Difference of M

Table 18: ADF test for the price (Mt) with intercept

 Null Hypothesis: M has a unit root Exogenous: Constant Lag Length: 9 (Automatic – based on SIC, maxlag=13) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.159116 0.2222 Test critical values: 1% level -3.468980 5% level -2.878413 10% level -2.575844 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(M) Method: Least Squares Date: 05/08/15   Time: 11:25 Sample (adjusted): 11 180 Included observations: 170 after adjustments Variable Coefficient Std. Error t-Statistic Prob. M(-1) -0.050680 0.023473 -2.159116 0.0323 D(M(-1)) 0.420691 0.077708 5.413717 0.0000 D(M(-2)) 0.125929 0.078906 1.595946 0.1125 D(M(-3)) 0.070178 0.079388 0.883983 0.3780 D(M(-4)) -0.592196 0.079298 -7.467989 0.0000 D(M(-5)) 0.325686 0.089884 3.623391 0.0004 D(M(-6)) 0.120719 0.079042 1.527282 0.1287 D(M(-7)) 0.007831 0.079588 0.098390 0.9217 D(M(-8)) -0.361934 0.079773 -4.537046 0.0000 D(M(-9)) 0.230074 0.081251 2.831650 0.0052 C 0.003824 0.001940 1.971436 0.0504 R-squared 0.395474 Mean dependent var -8.37E-05 Adjusted R-squared 0.357453 S.D. dependent var 0.010743 S.E. of regression 0.008611 Akaike info criterion -6.608934 Sum squared resid 0.011791 Schwarz criterion -6.406029 Log likelihood 572.7594 Hannan-Quinn criter. -6.526598 F-statistic 10.40159 Durbin-Watson stat 2.004225 Prob(F-statistic) 0.000000

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Table 19: ADF test for the first difference of the price (ΔMt) with intercept

 Null Hypothesis: M_1ST_DIFF has a unit root Exogenous: Constant Lag Length: 8 (Automatic – based on SIC, maxlag=13) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -4.851308 0.0001 Test critical values: 1% level -3.468980 5% level -2.878413 10% level -2.575844 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(M_1ST_DIFF) Method: Least Squares Date: 05/08/15   Time: 11:26 Sample (adjusted): 10 179 Included observations: 170 after adjustments Variable Coefficient Std. Error t-Statistic Prob. M_1ST_DIFF(-1) -0.887855 0.183013 -4.851308 0.0000 D(M_1ST_DIFF(-1)) 0.280723 0.162140 1.731367 0.0853 D(M_1ST_DIFF(-2)) 0.377803 0.153874 2.455275 0.0151 D(M_1ST_DIFF(-3)) 0.416141 0.146388 2.842715 0.0051 D(M_1ST_DIFF(-4)) -0.211577 0.135610 -1.560188 0.1207 D(M_1ST_DIFF(-5)) 0.088480 0.105686 0.837196 0.4037 D(M_1ST_DIFF(-6)) 0.192609 0.098897 1.947571 0.0532 D(M_1ST_DIFF(-7)) 0.179985 0.092115 1.953927 0.0525 D(M_1ST_DIFF(-8)) -0.205885 0.081390 -2.529601 0.0124 C -0.000113 0.000669 -0.169424 0.8657 R-squared 0.542757 Mean dependent var 4.83E-05 Adjusted R-squared 0.517038 S.D. dependent var 0.012532 S.E. of regression 0.008709 Akaike info criterion -6.591801 Sum squared resid 0.012137 Schwarz criterion -6.407342 Log likelihood 570.3031 Hannan-Quinn criter. -6.516950 F-statistic 21.10263 Durbin-Watson stat 1.992344 Prob(F-statistic) 0.000000

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Table 10: Augmented Dickey‐Fuller (ADF) test for residuals with two lagged changes and intercept

 Null Hypothesis: RESID01 has a unit root Exogenous: Constant Lag Length: 1 (Automatic – based on SIC, maxlag=2) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.776281 0.0637 Test critical values: 1% level -3.467205 5% level -2.877636 10% level -2.575430 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID01) Method: Least Squares Date: 05/08/15   Time: 11:32 Sample (adjusted): 3 180 Included observations: 178 after adjustments Variable Coefficient Std. Error t-Statistic Prob. RESID01(-1) -0.039417 0.014198 -2.776281 0.0061 D(RESID01(-1)) 0.517096 0.064276 8.044940 0.0000 C 6.65E-05 0.000306 0.217182 0.8283 R-squared 0.282030 Mean dependent var 0.000115 Adjusted R-squared 0.273825 S.D. dependent var 0.004791 S.E. of regression 0.004083 Akaike info criterion -8.147258 Sum squared resid 0.002917 Schwarz criterion -8.093633 Log likelihood 728.1060 Hannan-Quinn criter. -8.125512 F-statistic 34.37139 Durbin-Watson stat 2.166087 Prob(F-statistic) 0.000000 Econometrics Essay Help