For my project, I chose to evaluate the effect of Gross Domestic Income on personal consumption expenditures for alcoholic beverages purchased for off-site consumption between 1950 and 2015, at an annual frequency. Both data sets were set to an index where 1985 levels equaled 100. I chose GDI because it displays the aggregate income of the US population, and would compare well to total expenditures for alcoholic beverages, excluding commercial purchases and purchases in bars and restaurants. The results of my initial regression are below:
These regression results suggest that the level of Gross Domestic Income affects the level of for personal consumption expenditures for alcoholic beverages purchased for off-site consumption. For this sample, a one percentage increase in the index (1985 = 100) for Gross Domestic Income was associated with a subsequent 0.75658 percent increase in the index (1985 = 100) for personal consumption expenditures for alcoholic beverages purchased for off-site consumption- holding all else constant. The coefficient of personal consumption expenditures for alcoholic beverages purchased for off-site consumption was highly statistically significant (p<.0001). The regression fit very well (R^{2}=.9945, adjusted R^{2}=.9944) and was highly statistically significant (F=11522.8, p<.0001).
A Durban-Watson test revealed positive autocorrelation (DW=.1812, P<.0001). I ran the AUTOREG procedure with a 1 period lag to correct for first-order autocorrelation, and the corrected Durban-Watson test showed improved results using Yule-Walker estimates (DW=1.0787, p<.0001). The interpretation of the new corrected model is below:
These regression results suggest that the level of Gross Domestic Income affects the level of for personal consumption expenditures for alcoholic beverages purchased for off-site consumption. For this sample, a one percentage increase in the index (1985 = 100) for Gross Domestic Income was associated with a subsequent 0.7463 percent increase in the index (1985 = 100) for personal consumption expenditures for alcoholic beverages purchased for off-site consumption- holding all else constant. The coefficient of personal consumption expenditures for alcoholic beverages purchased for off-site consumption was highly statistically significant (p<.0001). The regression fit very well (R^{2}=.9549, total R^{2}=.9999)
SAS outputs are listed below:
The REG Procedure
Model: MODEL1
Dependent Variable: AlcC AlcC
Number of Observations Read | 66 |
Number of Observations Used | 66 |
Analysis of Variance | |||||
Source | DF | Sum of Squares |
Mean Square |
F Value | Pr > F |
Model | 1 | 620243 | 620243 | 11522.8 | <.0001 |
Error | 64 | 3444.94662 | 53.82729 | ||
Corrected Total | 65 | 623688 |
Root MSE | 7.33671 | R-Square | 0.9945 |
Dependent Mean | 111.81589 | Adj R-Sq | 0.9944 |
Coeff Var | 6.56142 |
Parameter Estimates | ||||||
Variable | Label | DF | Parameter Estimate |
Standard Error |
t Value | Pr > |t| |
Intercept | Intercept | 1 | 12.67278 | 1.29174 | 9.81 | <.0001 |
GDI | GDI | 1 | 0.75658 | 0.00705 | 107.34 | <.0001 |
The REG Procedure
Model: MODEL1
Dependent Variable: AlcC AlcC
Durbin-Watson D | 0.181 |
Pr < DW | <.0001 |
Pr > DW | 1.0000 |
Number of Observations | 66 |
1st Order Autocorrelation | 0.906 |
The AUTOREG Procedure
Dependent Variable | AlcC |
AlcC |
The AUTOREG Procedure
Ordinary Least Squares Estimates | |||
SSE | 3444.94662 | DFE | 64 |
MSE | 53.82729 | Root MSE | 7.33671 |
SBC | 456.709786 | AIC | 452.330477 |
MAE | 6.27164079 | AICC | 452.520953 |
MAPE | 14.4722973 | HQC | 454.06095 |
Durbin-Watson | 0.1812 | Regress R-Square | 0.9945 |
Total R-Square | 0.9945 |
Durbin-Watson Statistics | |||
Order | DW | Pr < DW | Pr > DW |
1 | 0.1812 | <.0001 | 1.0000 |
Parameter Estimates | ||||||
Variable | DF | Estimate | Standard Error |
t Value | Approx Pr > |t| |
Variable Label |
Intercept | 1 | 12.6728 | 1.2917 | 9.81 | <.0001 | |
GDI | 1 | 0.7566 | 0.007048 | 107.34 | <.0001 | GDI |
Estimates of Autocorrelations | |||
Lag | Covariance | Correlation | -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 |
0 | 52.1962 | 1.000000 | | |********************| |
1 | 47.2653 | 0.905531 | | |****************** | |
Preliminary MSE | 9.3960 |
Estimates of Autoregressive Parameters | |||
Lag | Coefficient | Standard Error |
t Value |
1 | -0.905531 | 0.053454 | -16.94 |
Corrected for Autocorrelation using Yule-Walker |
The AUTOREG Procedure
Yule-Walker Estimates | |||
SSE | 595.600715 | DFE | 63 |
MSE | 9.45398 | Root MSE | 3.07473 |
SBC | 346.778016 | AIC | 340.209052 |
MAE | 2.24931649 | AICC | 340.596149 |
MAPE | 3.10491344 | HQC | 342.804762 |
Durbin-Watson | 1.0787 | Regress R-Square | 0.9549 |
Total R-Square | 0.9990 |
Durbin-Watson Statistics | |||
Order | DW | Pr < DW | Pr > DW |
1 | 1.0787 | <.0001 | 1.0000 |
Parameter Estimates | ||||||
Variable | DF | Estimate | Standard Error |
t Value | Approx Pr > |t| |
Variable Label |
Intercept | 1 | 13.7445 | 4.6762 | 2.94 | 0.0046 | |
GDI | 1 | 0.7463 | 0.0204 | 36.52 | <.0001 | GDI |