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As one would expect, the greater the variability in a given variable the higher the elasticity, especially when the variables either measure purchasing power as pi does directly or how the variables stock, and index of consumer sentiment also are shown as a result of their large variances. Taking a step back from the statistical analysis and thinking logically about this, elasticity would be defined by the level of car stocks or inventories on hand, customer attitudes and behaviors and the amount of money they had to spend. These three variables delivered a 75% R2 correlation coefficient. Elasticity is a function of price and demand so these series of relationships make sense.
Forecasting Looking for those variables that define variability first is critical, hence the completion of a thorough correlation analysis. Following this, a stepwise regression was completed to show the combined effects of the most correlated variables, which illustrated the strong effects of personal income (pi), cars in stock (stock) and the implications of excess inventory driving down price, and consumer sentiment (sentiment) impacting elasticity of unitsales. Forecasting unitsales using exponential smoothing under the Winters Method proved the most reliable in defining the next four quarters of unitsales given the sample data.
The first step in completing a forecast is to define the confidence intervals. Both 90% and 95% confidence intervals are typically chosen, for purposes of this exercise the latter figure, 95% has been selected. A 95% confidence interval ensures statistically significant results and a higher level of reliability when applying the results. Presented below are the results of a one-tailed Z-Test and t-test for confidence intervals based on the dependent variable unitsales.
Confidence Interval for Mean using Z. from Infinite Population or with Replacement
Lower limit =
Upper Limit =
Margin for Error (Half Width) =
Sample Mean =
Standard Deviation =
Confidence Interval =
Sample Size =
Confidence Interval for Mean using t from Infinite Population or with Replacement
Lower limit =
Upper Limit =
Margin for Error (Half Width) =
Sample Mean =
Sample Standard Deviation =
Confidence Interval =
Sample Size =
The fact that both of these tests show minimal differences between Upper and Lower Limits indicates the data has slight variability, as is shown by the standard deviation being 1.1976. This shows little variability across the sample data.
Forecasting unit sales of automobiles can be accomplished using a wide variety of techniques. Using Winter's Model for exponential smoothing as interpreted by the KADD Microsoft Excel add-on statistical tool generates the following forecast for the variable unitsales.
UnitSales forecast for the next four quarters using the Winters Method of Exponent Smoothing from KADD using a smoothing constraint of.2 for alpha,.2 for beta,.2 for gamma, for four periods delivers the following forecast:
UnitSales (Projected for 2005)
Q1, 2005 11,453,000
Q2, 2005 11,290,200
Q3, 2005 11,203,200
Q4, 2005 10,772,300
Appendix a: Correlation Matrix of all variables
Unitsales Pi sentiment Unemr costcarown mpg strike cardeprate avgprice finrate Stock Pearson Correlation unitsales 1.000.624.005.176.422.361 -.304.244.512.248.541 Pi.624 1.000 -.530.592.924.578 -.257 -.262.938.808.976 sentiment.005 -.530 1.000 -.482 -.443.128.240.723 -.373 -.737 -.571 Unemr.176.592 -.482 1.000.654.285 -.201 -.618.642.614.738 Costcarown.422.924 -.443.654 1.000.744 -.216 -.275.986.793.939 Mpg.361.578.128.285.744 1.000 -.080.207.780.303.559 Strike -.304 -.257.240 -.201 -.216 -.080 1.000.045 -.227 -.122 -.266 cardeprate.244 -.262.723 -.618 -.275.207.045 1.000 -.229 -.498 -.396 avgprice.512.938 -.373.642.986.780 -.227 -.229 1.000.735.947 Finrate.248.808 -.737.614.793.303 -.122 -.498.735 1.000.820 Stock.541.976 -.571.738.939.559 -.266 -.396.947.820 1.000 Sig. (1-tailed) unitsales..000.484.089.000.002.009.030.000.028.000 Pi.000..000.000.000.000.024.022.000.000.000 sentiment.484.000..000.000.165.032.000.002.000.000 Unemr.089.000.000..000.014.062.000.000.000.000 Costcarown.000.000.000.000..000.049.017.000.000.000 Mpg.002.000.165.014.000..273.057.000.009.000 Strike.009.024.032.062.049.273..367.040.176.020 cardeprate.030.022.000.000.017.057.367..039.000.001 avgprice.000.000.002.000.000.000.040.039..000.000 Finrate.028.000.000.000.000.009.176.000.000..000 Stock.000.000.000.000.000.000.020.001.000.000.
Variable Name
Variable Description
Unitsales
Unit sales of automobiles. SAAR in millions of cars.
Pi
Personal Income less transfer payments. SAAR in billions of $
Sentiment
Index of consumer sentiment. 1980 Q1 = 100.
Unemr
Unemployment rate. Percent.
Costcarown
Index of cost of car ownership. 1987 = 1.0.
Mpg
Average miles per gallon of current model year cars.
Strike
Indicator variable for UAW strikes. 1=strike.
Cardeprate
Stock of cars depreciation rate.
Avgprice
Index of average price of a new car.
Stock
Stock of cars. Millions.
Finrate
Finance rate on automobile loans. Percent.
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