In analyzing the stock market, we sometimes use the model y = 0 + B1x + to relate y, the rate of return on

In analyzing the stock market, we sometimes use the model y = β0 + B1x + ε to relate y, the rate of return on a particular stock, to x, the rate of return on the overall stock market. When using the preceding model, we can interpret β1 to be the percentage point change in the mean (or expected) rate of return on the particular stock that is associated with an increase of one percentage point in the rate of return on the overall stock market.
If regression analysis can be used to conclude (at a high level of confidence) that /3, is greater than l (for example, if the 95 percent confidence interval for β1 were [l.1826, l.4723]). this indicates that the mean rate of return on the particular stock changes more quickly than the rate of return on the overall stock market. Such a stock is called an aggressive stock because gains for such a stock tend to be greater than overall market gains (which occur when the market is bullish). However, losses for such a stock tend to be greater than overall market losses (which occur when the market is bearish). Aggressive stocks should be purchased if you expect the market to rise and avoided if you expect the market to fall.
If regression analysis can be used to conclude (at a high level of confidence) that /3, is less than l (for example, if the 95 percent confidence interval for β1 were [.4729, .78611), this indicates that the mean rate of return on the particular stock changes more slowly than the rate of return on the overall stock market. Such a stock is called a defensive stock. Losses for such a stock tend to be less than overall market losses, whereas gains for such a stock tend to be less than overall market gains. Defensive stocks should be held if you expect the market to fall and sold off if you expect the market to rise.
If the least squares point estimate b1 of β1, is nearly equal to 1, and if the 95 percent confidence interval for β1 contains 1. this might indicate that the mean rate of return on the particular stock changes at roughly the same rate as the rate of return on the overall stock market. Such a stock is called a neutral stock.
In a 1984 article in Financial Analysts Journal. Haim Levy considers how a stock's value of β1, depends on the length of time for which the rate of return is calculated. Levy calculated estimated values of β1 for return length times varying from 1 to 30 months for each of 38 aggressive stocks, 38 defensive stocks, and 68 neutral stocks. Each estimated value was based on data from 1946 to 1975. In the following table we present the average estimate of β1 for each stock type for different return length times:

Let y = average estimate of β1, and x = return length time, and consider relating y to x for each stock type by using the simple linear regression model
y= β*0 + β*1x + e
Here β*0 and β*1 are regression parameters relating y to x. We use the asterisks to indicate that these regression parameters are different from β0 and β1. Calculate a 95 percent confidence interval for β*1 for each stock type. Carefully interpret the meaning of each interval.

Average Estimate of B Return Neutra Defensive Stocks 50 Aggressive Stocks 1.37 1.42 1.53 1.69 1.83 1.67 1.78 1.86 1.83 Length Time 41 39 40 38 39 35 .33 Stocks 98 95 94 1.00 98 1.00 1.02 1.14 1.22 12 15 18 24 30