Car Sales

Analysis of monthly sales of light weight vehicles.

Laura PomellaKaren ChangHeidi BraungerDavid ParkerDerek ShumMike Hu

Car Sales Overview

Slight trend in mean and variance.

Car Sales Overview (2)

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Series: CARSALESSample 1976:01 2006:04Observations 364

Mean 14.62159Median 14.80000Maximum 21.70000Minimum 8.850000Std. Dev. 2.094847Skewness -0.308770Kurtosis 3.401570

Jarque-Bera 8.229632Probability 0.016329

Histogram of Car Sales

Trend?Dependent Variable: CARSALES

Method: Least Squares

Date: 06/01/06 Time: 19:23

Sample: 1976:01 2006:04

Included observations: 364

Variable Coefficient Std. Error t-Statistic Prob.

TREND 0.012735 0.000804 15.83336 0.0000

C 12.29747 0.169378 72.60349 0.0000

R-squared 0.409168 Mean dependent var 14.62159

Adjusted R-squared 0.407536 S.D. dependent var 2.094847

S.E. of regression 1.612439 Akaike info criterion 3.798853

Sum squared resid 941.1859 Schwarz criterion 3.820266

Log likelihood -689.3912 F-statistic 250.6953

Durbin-Watson stat 0.449535 Prob(F-statistic) 0.000000

Correlogram of CARSALES

• Large spike in autocorrelation function at lag one equal to 0.864.

• Slow decay in ACF suggesting stationary series.

• Looked to Unit-Root Test for confirmation.

Unit-Root Test (2 Lags)

Unit-Root Test (3 Lags)

Pre-Whitened Series

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DCARSALES

Heteroscedasticity?

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DCARSALES

Heteroscedasticity?

Histogram of DCARSALES

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Series: DCARSALESSample 1976:02 2006:04Observations 363

Mean 0.011515Median 0.060000Maximum 5.650000Minimum -6.700000Std. Dev. 1.081097Skewness -0.873618Kurtosis 12.42232

Jarque-Bera 1388.972Probability 0.000000

• Single-peaked and very kurtotic suggesting conditional heteroscedasticity.

Correlogram of DCARSALES

Unit-Root Test (1 Lag)

First-Differenced Log

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DLNCARSALES

Did nothing to help with any trend in the variance.

DCARSALES as ARMA(2,1)

ARMA(2,1) Residuals

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Residual Actual Fitted

ARMA(2,1) Residuals (2)

ARMA(2,1) Residuals (3)

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Series: ResidualsSample 1976:04 2006:04Observations 361

Mean -0.000427Median -0.005453Maximum 5.326101Minimum -5.077401Std. Dev. 0.931135Skewness 0.281574Kurtosis 10.95220

Jarque-Bera 955.9681Probability 0.000000

Single-peaked, non-normal and very kurtotic suggesting conditional heteroscedasticity.

Breusch-Godfrey Test

Competing Model - ARMA(2,2)

ARMA(2,2) Residuals

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Residual Actual Fitted

ARMA(2,2) Residuals (2)

ARMA(2,2) Residuals (3)

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Series: ResidualsSample 1976:04 2006:04Observations 361

Mean 0.000940Median 0.008480Maximum 5.305898Minimum -5.196835Std. Dev. 0.928319Skewness 0.332320Kurtosis 11.36485

Jarque-Bera 1059.122Probability 0.000000

Single-peaked and very kurtotic suggesting conditional heteroscedasticity.

Breusch-Godfrey Test – ARMA(2,2)

ARCH TestARCH Test:

F-statistic 15.90085 Probability 0.000081

Obs*R-squared 15.30969 Probability 0.000091

Test Equation:

Dependent Variable: RESID^2

Method: Least Squares

Date: 06/01/06 Time: 19:47

Sample(adjusted): 1976:05 2006:04

Included observations: 360 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C 0.684056 0.150044 4.559023 0.0000

RESID^2(-1) 0.206220 0.051715 3.987587 0.0001

R-squared 0.042527 Mean dependent var 0.861758

Adjusted R-squared 0.039852 S.D. dependent var 2.774273

S.E. of regression 2.718431 Akaike info criterion 4.843526

Sum squared resid 2645.572 Schwarz criterion 4.865116

Log likelihood -869.8348 F-statistic 15.90085

Durbin-Watson stat 1.993423 Prob(F-statistic) 0.000081

GARCH(1,1) with ARMA(2,2)

GARCH(1,1) Residuals

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GARCH(1,1) Residuals (2)

GARCH(1,1) Residuals (3)

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Series: Standardized ResidualsSample 1976:04 2006:04Observations 361

Mean -0.036544Median -0.088254Maximum 7.083473Minimum -3.291765Std. Dev. 1.000357Skewness 1.194297Kurtosis 10.27127

Jarque-Bera 881.0919Probability 0.000000

Single-peaked but slightly skewed, non-normal and very kurtotic.

GARCH(1,1) Residuals Squared

ARCH Test

In-Sample Forecast

DATE Forecast dcarsales SEF H(t)

5/1/2005 -0.54 -0.05354 0.621858 0.386707

6/1/2005 1.18 -0.06959 0.818547 0.568173

7/1/2005 2.86 -0.00681 0.938083 0.706278

8/1/2005 -3.93 0.044427 1.016274 0.811383

9/1/2005 -0.41 0.060687 1.073035 0.891373

10/1/2005 -1.64 0.055308 1.114609 0.95225

11/1/2005 1 0.045907 1.145115 0.99858

12/1/2005 1.45 0.040856 1.167793 1.03384

1/1/2006 0.43 0.040334 1.184778 1.060674

2/1/2006 -1.03 0.041654 1.197543 1.081096

3/1/2006 -0.01 0.042793 1.207166 1.096639

4/1/2006 0.15 0.04318 1.214438 1.108468

In-Sample Forecast Plots• Forecast and confidence interval for months June 2005 to April 2006.

In-Sample Forecast Plots (2)

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DCARSALESDCARSHORT F

UPPERLIMITLOWERLIMIT

Forecast Through April 2007

Forecast Through April 2007 (2)

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DCARSALESFORECAST

UPPER_INT ERVALLOWER_INT ERVAL

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DCARSALESFORECAST

FORECAST +2*SEFFORECAST -2*SEF

Recolored!

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FORECASTCARSALES

UPPER_LIMITLOWER_LIMIT

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FORECASTCARSALES

UPPER_LIMITLOWER_LIMIT

G o t C a r ?