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

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