In Problem 14.4 on page 583, you used sales and number of...

In Problem 14.4 on page 583, you used sales and number of orders to predict distribution costs at a mail-order catalog business (stored in WareCost). a. Plot the residuals versus $\hat{Y}_i$. b. Plot the residuals versus $X_{1 i}$. c. Plot the residuals versus $X_{2 i}$. d. Plot the residuals versus time. e. In the residual plots created in (a) through (d), is there any evidence of a violation of the regression assumptions? Explain. f. Determine the Durbin-Watson statistic. g. At the 0.05 level of significance, is there evidence of positive autocorrelation in the residuals? data?

In Problem 14.4 on page 583, you used sales and number of orders to predict distribution costs at a mail-order catalog business (stored in WareCost).
a. Plot the residuals versus $\hat{Y}_i$.
b. Plot the residuals versus $X_{1 i}$.
c. Plot the residuals versus $X_{2 i}$.
d. Plot the residuals versus time.
e. In the residual plots created in (a) through (d), is there any evidence of a violation of the regression assumptions? Explain.
f. Determine the Durbin-Watson statistic.
g. At the 0.05 level of significance, is there evidence of positive autocorrelation in the residuals?
data?

Key Concepts

Residual Plots

Residual plots are graphical tools used to assess the appropriateness of a regression model by plotting the residuals (the differences between observed and predicted values) against various variables such as fitted values, predictor variables, or time. These plots help identify patterns, trends, non-linearity, heteroscedasticity, or outliers that suggest potential issues with the model fit.

Regression Assumptions

Regression analysis relies on several key assumptions such as linearity, independence, homoscedasticity (constant variance), and normality of residuals. The evaluation of these assumptions is crucial because violations can lead to biased estimates, unreliable significance tests, and incorrect inference about relationships between variables.

Graphical Diagnostics

Graphical diagnostic techniques involve using plots, such as residual versus fitted values or residual versus individual predictors, to visually detect departures from the model assumptions. These techniques complement statistical tests by providing a visual representation of potential anomalies like trends, cycles, or structure in the residuals.

Autocorrelation

Autocorrelation refers to the correlation of a time series with its own past values. In the context of regression residuals, autocorrelation indicates that the residuals are not independent over time, which can invalidate hypothesis tests and inference if the issue is not addressed. It is particularly a concern in time series regression analysis.

Durbin-Watson Statistic

The Durbin-Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals of a regression model. Values close to 2 suggest no autocorrelation, while values deviating significantly from 2 provide evidence of positive or negative autocorrelation, thus helping to evaluate the independence assumption in regression analysis.

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