For This Assignment, Complete Problem 64

For This Assignment You Are Required To Complete Problem 64 P 593

For this assignment, you are required to complete Problem 64 (p. 593) in Chapter 12 of your textbook. Use the appropriate Excel file templates from. Once complete, post your Excel Document in Waypoint. Show all work.

23-64) Let Yt be the sales during month t (in thousands of dollars) for a photography studio, and let Pt be the price charged for portraits during month t. The data are in the file Week 4 Assignment Chapter 12 Problem 64. Use regression to fit the following model to these data: Yt = a + b1Yt−1 + b2Pt + et. This equation indicates that last month’s sales and the current month’s price are explanatory variables. The last term, et, is an error term. If the price of a portrait during month 21 is $10, what would you predict for sales in month 21? Does there appear to be a problem with autocorrelation of the residual? Explain your answer.

Paper For Above instruction

Effective analysis of time-series data, such as monthly sales for a photography studio, requires a comprehensive understanding of regression models, including potential issues such as autocorrelation. In this context, we analyze how past sales and current prices influence current sales and whether the residuals from the regression exhibit autocorrelation, which can jeopardize the validity of the model's inferences.

The specified model, Yt = a + b1Yt−1 + b2Pt + et, is a multiple linear regression where Yt denotes sales in month t, Yt−1 the sales from the previous month, and Pt the current month’s price. Including a lagged dependent variable (Yt−1) captures potential inertia or momentum in sales, while the inclusion of current price (Pt) accounts for price sensitivity. The error term, et, embodies unobserved influences on sales that are not captured by the model.

To conduct this analysis, one would typically use Excel's regression tools or similar statistical software. Initially, data from the provided Excel file should be imported, with variables arranged properly for regression analysis. The model parameters, a, b1, and b2, are estimated through least squares regression, which minimizes the sum of squared residuals. Once the regression output is obtained, the coefficients can be used to predict sales for month 21, particularly if the previous month's sales and current price are known.

Given a price of $10 for month 21, and assuming the data had estimated parameters a, b1, and b2, the predicted sales can be calculated as:

Ŷ₍₂₁₎ = a + b1 Y₍₂₀₎ + b2 10

where Y₍₂₀₎ represents the actual sales in month 20. The specific value of Y₍₂₀₎ should be extracted from the dataset, and plugging in the estimated coefficients yields the forecasted sales.

Assessing the presence of autocorrelation in residuals is essential because autocorrelation indicates that residuals are correlated across time, violating a key assumption of classical regression. It can be detected through residual plots, the Durbin-Watson test, or autocorrelation function (ACF) plots. If residuals exhibit autocorrelation, the reliability of hypothesis tests and confidence intervals for the estimated coefficients diminishes, leading to misleading inferences about the relationships between sales, past sales, and prices.

Typically, a Durbin-Watson statistic close to 2 suggests no autocorrelation; values significantly less than 2 indicate positive autocorrelation, while values greater than 2 suggest negative autocorrelation. If autocorrelation is detected, remedial measures include model modifications or using time series specific methods such as ARIMA models or adding lagged residuals as predictors.

In conclusion, this exercise illustrates how regression modeling, combined with residual diagnostics, provides insights into the factors driving sales and the adequacy of the model. Addressing autocorrelation is crucial for ensuring the robustness of forecast accuracy and inference validity, especially in time-series data analysis.

References

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