Given the opportunity  to model an Engel Curve of my choice, I modeled the dependent variable of real personal consumption expenditures of musical instruments with real disposable personal income all in billions of chained 2009 US dollars. The purpose of using these specific data sets is to compare the entirety of US expenditures of musical instruments to the entirety of real disposable personal income since it isn’t possible to find the average number of musical instruments per household or per capita. The assumption prior to regressing the model in SAS is to assume that the relationship is positive since musical instruments are commonly thought of as luxury goods.

Using Ordinary Least Squares in SAS produced these results:
The slope coefficient for Income is statistically significant at the 5% level with a p-level of 0.0440. The value of the slope coefficient was 0.00019, indicating that an increase in 1 dollar of income increases consumption of musical instruments by 0.00019 dollars. Or, an increase in 100,000 dollars in income increases consumption in musical instruments by 19 dollars.
The model, however, was a mildly good fit of the data as the R-squared value was 0.244, meaning that the model explains 24.3% of the variance in the data. And given the lowered adjusted R-squared value of 0.193 indicates that the variable of income may not a good explanatory variable for the consumption of musical instruments.

Money and Music

The REG Procedure
Model: MODEL1
Dependent Variable: Music Music
Number of Observations Read 17
Number of Observations Used 17

Analysis of Variance
Source DF Sum of
Squares Mean
Square F Value Pr > F
Model 1 0.73709 0.73709 4.83 0.0440
Error 15 2.28761 0.15251
Corrected Total 16 3.02471

Root MSE 0.39052 R-Square 0.2437
Dependent Mean 4.98235 Adj R-Sq 0.1933
Coeff Var 7.83811

Parameter Estimates
Variable Label DF Parameter
Estimate Standard
Error t Value Pr > |t|
Intercept Intercept 1 2.96061 0.92449 3.20 0.0059
Income Income 1 0.00019190 0.00008729 2.20 0.0440

Because time-series data was used, the model risks experiencing the problem of autocorrelation. A Durbin Watson Test was conducted and the value provided by SAS dictated that the model was at risk with a statistic of 0.6404 and a p-value of 0.0001.
By lagging the model and analyzing Yule-Walker estimates recommended by SAS, a new slope coefficient for Income was produced at a level still significant at the 5% level: 0.0139. The coefficient itself was 0.000326, meaning that a 1 dollar increase in income, increases expenditure in musical instruments by 0.000326 dollars. Also, the lagged model gave a new Regressed R-squared value of 0.3608 and a total R-squared value of 0.5618. Analyzing the Regressed R-squared value, which is greater than the value offered in the non-lagged model, the model properly fits 36.1% of the data.

Durbin-Watson Statistics
Order DW Pr < DW Pr > DW
1 0.6404 0.0001 0.9999
The AUTOREG Procedure
Ordinary Least Squares Estimates
SSE 2.2876127 DFE 15
MSE 0.15251 Root MSE 0.39052
SBC 19.8133593 AIC 18.1469326
MAE 0.3167016 AICC 19.0040755
MAPE 6.55221092 HQC 18.3125787
Durbin-Watson 0.6404 Regress R-Square 0.2437
Total R-Square 0.2437

Parameter Estimates
Variable DF Estimate Standard
Error t Value Approx
Pr > |t| Variable Label
Intercept 1 2.9606 0.9245 3.20 0.0059
Income 1 0.000192 0.0000873 2.20 0.0440 Income

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 0.1346 1.000000 | |********************|
1 0.0711 0.528232 | |*********** |

Preliminary MSE 0.0970

Estimates of Autoregressive Parameters
Lag Coefficient Standard
Error t Value
1 -0.528232 0.226932 -2.33
Money and Music

The AUTOREG Procedure
Yule-Walker Estimates
SSE 1.32531823 DFE 14
MSE 0.09467 Root MSE 0.30768
SBC 13.6941736 AIC 11.1945335
MAE 0.24502541 AICC 13.0406874
MAPE 5.0736045 HQC 11.4430027
Durbin-Watson 1.2817 Regress R-Square 0.3608
Total R-Square 0.5618

Parameter Estimates
Variable DF Estimate Standard
Error t Value Approx
Pr > |t| Variable Label
Intercept 1 1.5193 1.2293 1.24 0.2368
Income 1 0.000326 0.000116 2.81 0.0139 Income

A test for heteroskedasticity was originally considered, but given the small sample size and a look at scatter plot data for the dependent and explanatory variable, there wasn’t any real sign that the model might suffer from a second standard regression problem.

Seeing the results of the model and even the scatter plot above indicates that the original assumption that musical instruments and income have a positive relationship and that musical instruments are a luxury good have been confirmed. While the effect of income on musical instrument consumption and the R-squared value are low, the model still gives a good idea of what the Engel Curve is.

The Hills are Alive with the Sound of Music(al Instruments).
Tagged on: