The data set below includes observed high temperatures for $137$ consecutive days in a$$ city, along with predictions of this temperature made$1,2,3,4,5,6,$ and $7$ days before by a local weather station that advertises its ability to predict weather a week ahead. The data set also includes the$30-$year average high temperature for each day. Complete parts$($a$)$ through$($f$).$

Summarize the simple regression of the high temperature on the$30-$year average high temperature.

The simple regression equation is ModifyingAbove y with caretequals

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b$)$Explain why the simple regression estimated in$($a$)$ does not meet the conditions required for the SRM$.$

Select the correct choice and fill in the answer box to complete your choice.

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A$.$

There is a significant autocorrelation. The value of rho Subscript e is

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B$.$

There is multicollinearity. The VIF is

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C$.$

The slope is not significantly different from zero. The test statistic is

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C$)$o improve the predictions from the simple$$ regression, add all of the other explanatory variables provided by the local weather forecasts. What problem does the addition of these seven variables$$ introduce? What problem is$$ resolved?

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$1-$Day Ahead Forecast

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$2-$Day Ahead Forecast

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$3-$Day Ahead Forecast

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$4-$Day Ahead Forecast

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$5-$Day Ahead Forecast

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$6-$Day Ahead Forecast

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$7-$Day Ahead Forecast

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$$

Average High

D$)$What problem does the addition of these seven variables$$ introduce? Select the correct choice and fill in the answer box to complete your choice.

$($Round to two decimal places as$$ needed.$)$

A$.$

There is a significant autocorrelation. The value of rho Subscript e is

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B$.$

None of the regression coefficients are significantly different from zero. The$$ F$-$test statistic is

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C$.$

There is high collinearity. The largest VIF is

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Part $5$

What problem is$$ resolved? Select the correct choice and fill in the answer box to complete your choice.

$($Round to two decimal places as$$ needed.$)$

A$.$

Unlike in the$$ SRM$,$ the overall regression is now significant. The$$ F$-$test statistic is

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B$.$

Unlike in the$$ SRM$,$ there is no longer any high collinearity. The largest VIF is

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C$.$

Unlike in the$$ SRM$,$ there is no longer a significant autocorrelation. The value of rho Subscript e is

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Part $6$

$($d$)$ Does the estimated model satisfy the conditions of the$$ MRM$?$

A$.$

$$Yes$,$ but the presence of high collinearity indicates that the model could be improved by removing correlated variables.

B$.$

$$Yes$,$ because the$$ F$-$test indicates that at least one of the slopes is significantly different from zero.

C$.$

$$No$,$ because there is a significant autocorrelation.

D$.$

$$No$,$ because none of the slopes are significantly different from zero.

Part $7$

$($e$)$ Improve the model estimated in$($c$)$ and summarize a regression that you believe offers better predictions of the high temperature than the regression estimated using all of the explanatory variables.

Remove explanatory variables with small$$ t$-$statistics from the full model one at a time. If several variables have small$$ t$-$statistics, remove the variable with the highest VIF. Repeat this process until all regression coefficients are significantly different from zero. Enter $0$ into the answer boxes for any coefficients that are not included in the final model. Use alphaequals$0\mathrm{.}05.$

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$1-$Day Ahead Forecast

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$2-$Day Ahead Forecast

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$3-$Day Ahead Forecast

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$4-$Day Ahead Forecast

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$5-$Day Ahead Forecast

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$6-$Day Ahead Forecast

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$7-$Day Ahead Forecast

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$$

Average High

$($Round to three decimal places as$$ needed.$)$