For my SAS assignment, I wanted to test the effects of income on fine, collected commodities. Originally, I wanted to test incomes effect on fine art consumption, but was unable to find the appropriate data on art expenditure. Therefore, I decided to test disposable personal income and expenditure on jewelry and watches – expensive and often collected objects that are ‘cousins’ of fine art.
Below are two graphs showing the change in disposable personal income from 1970-2014 and the change in personal expenditure on jewelry and watches from 1970-2014. With these graphs in mind, and given their positive slope, there is a positive relationship between disposable income and expenditure on fine objects such as jewelry and watches. With this insight in mind, it is safe to hypothesize that jewelry and watches are normal goods – such that their demand increases as income increases. Throughout this paper, I will place the two measure in context with each other and improve upon the models limitations to create a superior model that may provide additional insight into the relationship between the two measures.
The MEANS Procedure
Variable | Label | N | Mean | Std Dev | Minimum | Maximum |
DISPINC
EXPONJW |
DISPINC
EXPONJW |
46
46 |
7325.30
36.2413043 |
2793.54
22.9113904 |
3413.10
4.1000000 |
12343.20
77.2000000 |
I included this table to provide descriptive statistics for the variables. These statistics show that the mean disposable income from 1970-2014 was $7,325.3 billion and the mean for the expenditure on jewelry and watches was $36.24 billion. Additionally, these results show the strong variance among the values over the almost 45 years of data. With a standard deviation of $2,793.54 billion for the DISPINC variable and $22.91 billion for the EXPONJW variable, the DISPINC variable had much more spread out values compared to the EXPONJW variable. With these statistics about the variables in mind, it is important to run OLS for the model.
I began by running OLS for the variables and testing for multicollinearity within the model. Each of the variables are measured in billions of dollars.
Personal Expenditure on Luxury Items – Jewelry and Watches as Disposable Personal Income changes
The REG Procedure
Model: MODEL1
Dependent Variable: EXPONJW EXPONJW
Number of Observations Read | 47 |
Number of Observations Used | 46 |
Number of Observations with Missing Values | 1 |
Analysis of Variance | |||||
Source | DF | Sum of Squares |
Mean Square |
F Value | Pr > F |
Model | 1 | 23425 | 23425 | 5228.10 | <.0001 |
Error | 44 | 197.14432 | 4.48055 | ||
Corrected Total | 45 | 23622 |
Root MSE | 2.11673 | R-Square | 0.9917 |
Dependent Mean | 36.24130 | Adj R-Sq | 0.9915 |
Coeff Var | 5.84066 |
Parameter Estimates | |||||||
Variable | Label | DF | Parameter Estimate |
Standard Error |
t Value | Pr > |t| | Variance Inflation |
Intercept | Intercept | 1 | -23.58639 | 0.88433 | -26.67 | <.0001 | 0 |
DISPINC | DISPINC | 1 | 0.00817 | 0.00011295 | 72.31 | <.0001 | 1.00000 |
My results are shown above, and there are no signs of multicollinearity since our VIF factor was below 5.
These regression results suggest that the level of disposable personal income affects the personal expenditure on jewelry and watches. For this sample, an increase in $1 billion of disposable personal income was associated with an increase of $0.00817 in personal expenditure on jewelry and watches, other things constant. The coefficient on the DISPINC variable was highly statistically significant (p<.0001). The regression as a whole fit very well (R^{2}=0.9917, adjusted R^{2}=0.9915) and was highly statistically significant (F = 5228.10, p=<.0001).
With these results from OLS, I suggest that, with not much surprise, jewelry and watches are normal goods and their demand increases as income increases.
Now, given the relative simplicity of the model and the possibility of autocorrelation, a test was performed to test for autocorrelation.
Personal Expenditure on Luxury Items – Jewelry and Watches as Disposable Personal Income changes
The REG Procedure
Model: MODEL1
Dependent Variable: EXPONJW EXPONJW
Number of Observations Read | 47 |
Number of Observations Used | 46 |
Number of Observations with Missing Values | 1 |
Analysis of Variance | |||||
Source | DF | Sum of Squares |
Mean Square |
F Value | Pr > F |
Model | 1 | 23425 | 23425 | 5228.10 | <.0001 |
Error | 44 | 197.14432 | 4.48055 | ||
Corrected Total | 45 | 23622 |
Root MSE | 2.11673 | R-Square | 0.9917 |
Dependent Mean | 36.24130 | Adj R-Sq | 0.9915 |
Coeff Var | 5.84066 |
Parameter Estimates | ||||||
Variable | Label | DF | Parameter Estimate |
Standard Error |
t Value | Pr > |t| |
Intercept | Intercept | 1 | -23.58639 | 0.88433 | -26.67 | <.0001 |
DISPINC | DISPINC | 1 | 0.00817 | 0.00011295 | 72.31 | <.0001 |
Personal Expenditure on Luxury Items – Jewelry and Watches as Disposable Personal Income changes
The REG Procedure
Model: MODEL1
Dependent Variable: EXPONJW EXPONJW
Durbin-Watson D | 0.682 |
Pr < DW | <.0001 |
Pr > DW | 1.0000 |
Number of Observations | 46 |
1st Order Autocorrelation | 0.659 |
Note:Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation.
For autocorrelation, my p=<.0001 suggests that there is positive autocorrelation present in the regression. In addition to autocorrelation, and something I will attend to at the same time as fixing autocorrelation, is the simplicity of the model. Later, I added the GDP variable to further understand personal expenditure on jewelry and watches.
I did not test or correct for heteroscedasticity since it usually appears in cross sectional data and large samples. Since this data set is a time series, it seems unnecessary to consider the possibility of heteroscedasticity.
To correct for autocorrelation and fix the simplicity of the model, I used the NLAG=1 command and added the GDP variable. This variable made the model more complex and allowed for relationships to be explained further.
Personal Expenditure on Luxury Items – Jewelry and Watches as Disposable Personal Income changes
The AUTOREG Procedure
Dependent Variable | EXPONJW |
EXPONJW |
Personal Expenditure on Luxury Items – Jewelry and Watches as Disposable Personal Income changes
The AUTOREG Procedure
Ordinary Least Squares Estimates | |||
SSE | 179.032184 | DFE | 43 |
MSE | 4.16354 | Root MSE | 2.04048 |
SBC | 204.538782 | AIC | 199.052858 |
MAE | 1.53619058 | AICC | 199.624286 |
MAPE | 5.96684571 | HQC | 201.107918 |
Durbin-Watson | 0.5743 | Total R-Square | 0.9924 |
Durbin-Watson Statistics | |||
Order | DW | Pr < DW | Pr > DW |
1 | 0.5743 | <.0001 | 1.0000 |
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation.
Parameter Estimates | ||||||
Variable | DF | Estimate | Standard Error |
t Value | Approx Pr > |t| |
Variable Label |
Intercept | 1 | -24.4985 | 0.9581 | -25.57 | <.0001 | |
DISPINC | 1 | 0.004443 | 0.001789 | 2.48 | 0.0170 | DISPINC |
GDP | 1 | 0.002801 | 0.001343 | 2.09 | 0.0430 | GDP |
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 | 3.8920 | 1.000000 | | |********************| |
1 | 2.7645 | 0.710315 | | |************** | |
Preliminary MSE | 1.9283 |
Estimates of Autoregressive Parameters | |||
Lag | Coefficient | Standard Error |
t Value |
1 | -0.710315 | 0.108612 | -6.54 |
Personal Expenditure on Luxury Items – Jewelry and Watches as Disposable Personal Income changes
The AUTOREG Procedure
Yule-Walker Estimates | |||
SSE | 78.4436822 | DFE | 42 |
MSE | 1.86771 | Root MSE | 1.36664 |
SBC | 171.111214 | AIC | 163.796648 |
MAE | 0.96082092 | AICC | 164.772258 |
MAPE | 4.11743445 | HQC | 166.536728 |
Durbin-Watson | 1.2994 | Transformed Regression R-Square | 0.9713 |
Total R-Square | 0.9967 |
Durbin-Watson Statistics | |||
Order | DW | Pr < DW | Pr > DW |
1 | 1.2994 | 0.0041 | 0.9959 |
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation.
Parameter Estimates | ||||||
Variable | DF | Estimate | Standard Error |
t Value | Approx Pr > |t| |
Variable Label |
Intercept | 1 | -25.7031 | 1.8207 | -14.12 | <.0001 | |
DISPINC | 1 | 0.000112 | 0.001904 | 0.06 | 0.9534 | DISPINC |
GDP | 1 | 0.006085 | 0.001435 | 4.24 | 0.0001 | GDP |
Above are my results from the correction of autocorrelation and the addition of the variable. Without knowing the functional form of the error variance, it is hard to say if correcting for autocorrelation is better than regular OLS. Overall, my original predictions were correct, and it seems that the addition of the GDP variable may have complicated matters and made the model worse-off. The DISPINC variable became an insignificant part of the model – although, it is an important part of the model in the original model. The intercept value of -$25.70 billion is appropriate since when disposable personal income is zero, there will be no expenditure on jewelry and watches, but rather the possibility of selling them to gain income needed for necessities. This negative intercept is also in the original model. And, the GDP variable became statistically significant. However, GDP and income are related, so that may account for this shift.
This model in the end does show that jewelry and watches are normal goods. My original model seemed more appropriate and fitting than the second mode. This may be due to the fact that I added a variable to make the model better, and inadvertently made it worse. If I had not added the GDP variable, but still corrected for autocorrelation these would be my results:
Personal Expenditure on Luxury Items – Jewerly and Watches as Disposable Personal Income changes
The AUTOREG Procedure
Ordinary Least Squares Estimates | |||
SSE | 197.144321 | DFE | 44 |
MSE | 4.48055 | Root MSE | 2.11673 |
SBC | 205.143182 | AIC | 201.485899 |
MAE | 1.55659992 | AICC | 201.764969 |
MAPE | 6.97946272 | HQC | 202.855939 |
Durbin-Watson | 0.6822 | Total R-Square | 0.9917 |
Durbin-Watson Statistics | |||
Order | DW | Pr < DW | Pr > DW |
1 | 0.6822 | <.0001 | 1.0000 |
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation.
Parameter Estimates | ||||||
Variable | DF | Estimate | Standard Error |
t Value | Approx Pr > |t| |
Variable Label |
Intercept | 1 | -23.5864 | 0.8843 | -26.67 | <.0001 | |
DISPINC | 1 | 0.008167 | 0.000113 | 72.31 | <.0001 | DISPINC |
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 | 4.2857 | 1.000000 | | |********************| |
1 | 2.8236 | 0.658828 | | |************* | |
Preliminary MSE | 2.4255 |
Estimates of Autoregressive Parameters | |||
Lag | Coefficient | Standard Error |
t Value |
1 | -0.658828 | 0.114724 | -5.74 |
Personal Expenditure on Luxury Items – Jewerly and Watches as Disposable Personal Income changes
The AUTOREG Procedure
Yule-Walker Estimates | |||
SSE | 111.528164 | DFE | 43 |
MSE | 2.59368 | Root MSE | 1.61049 |
SBC | 183.33677 | AIC | 177.850846 |
MAE | 1.11924553 | AICC | 178.422274 |
MAPE | 4.50892955 | HQC | 179.905906 |
Durbin-Watson | 1.3970 | Transformed Regression R-Square | 0.9685 |
Total R-Square | 0.9953 |
Durbin-Watson Statistics | |||
Order | DW | Pr < DW | Pr > DW |
1 | 1.3970 | 0.0119 | 0.9881 |
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation.
Parameter Estimates | ||||||
Variable | DF | Estimate | Standard Error |
t Value | Approx Pr > |t| |
Variable Label |
Intercept | 1 | -23.4209 | 1.7817 | -13.15 | <.0001 | |
DISPINC | 1 | 0.008144 | 0.000224 | 36.33 | <.0001 | DISPINC |
In the end, the models are not entirely different. However, there is a clear relationship between success, both in the economy and in personal income, and increases in expenditure on jewelry and watches. These are fine objects that are valuable goods that individuals often collect and present as gifts. Given these results, jewelry and watches are normal goods and their relative demands increase as successes are felt throughout the economy.