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 (R2=0.9917, adjusted R2=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.

The relationship between Personal expenditure on Jewelry and Watches and Disposable Personal Income 1970-2014