The stock market Some people think that the behavior of the...

The stock market Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year. Take the explanatory variable $x$ to be the percent change in a stock market index in January and the response variable $y$ to be the change in the index for the entire year. We expect a positive correlation between $x$ and $y$ because the change during January contributes to the full year's change. Calculation from data for an 18 -year period gives $$\begin{array}{c}\bar{x}=1.75 \% \quad s_{x}=5.36 \% \quad \bar{y}=9.07 \% \\ s_{y}=15.35 \% \quad r=0.596\end{array}$$ (a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work. (b) Suppose that the percent change in a particular January was 2 standard deviations above average. Predict the percent change for the entire year, without using the least-squares line. Show your work.

The stock market Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year. Take the explanatory variable $x$ to be the percent change in a stock market index in January and the response variable $y$ to be the change in the index for the entire year. We expect a positive correlation between $x$ and $y$ because the change during January contributes to the full year's change. Calculation from data for an 18 -year period gives
$$\begin{array}{c}\bar{x}=1.75 \% \quad s_{x}=5.36 \% \quad \bar{y}=9.07 \% \\
s_{y}=15.35 \% \quad r=0.596\end{array}$$
(a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work.
(b) Suppose that the percent change in a particular January was 2 standard deviations above average. Predict the percent change for the entire year, without using the least-squares line. Show your work.

Key Concepts

Least?Squares Regression

The least-squares regression method is used to determine the line that best fits a set of data points by minimizing the sum of the squared differences between the observed values and the values predicted by the line. The resulting regression line is typically expressed in slope-intercept form, where the slope is calculated as the product of the correlation coefficient and the ratio of the standard deviations of the response and explanatory variables, and the intercept is obtained by subtracting the product of the slope and the mean of the explanatory variable from the mean of the response variable.

Correlation Coefficient

The correlation coefficient measures the strength and direction of the linear relationship between two variables. Its value ranges from -1 to 1, with a positive value indicating a direct relationship and a negative value indicating an inverse relationship. This coefficient is essential in the context of regression because it not only tells us about the association between the variables but is also used in computing the slope of the regression line.

Standardization for Prediction

Standardization refers to the process of converting a raw score into a z-score which represents the number of standard deviations the score is away from the mean. This technique is useful in prediction scenarios where a value of the explanatory variable is given in terms of its deviation from the mean, allowing the analyst to estimate the corresponding change in the response variable by scaling the deviation appropriately using the ratio of the standard deviations and the correlation coefficient.

Explanatory and Response Variables

In regression analysis, the explanatory (or independent) variable is the factor assumed to influence changes in another variable, while the response (or dependent) variable is what is measured or predicted. This distinction is crucial because the objective is often to model and predict the response variable based on the observed values of the explanatory variable.

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