Econometrics

  • 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:

Econometrics2 Econometrics1

  • 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      
         
         

 

 

 

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      
         
         

 

 

 

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      
         
         

 

 

 

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      
         
         

 

 

 

  • 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      
         
         

 

 

 

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      
         
         

 

 

 

 

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      
         
         

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