Posts tagged “financial analysts”

- Eric Bank

Prediction interval

We continue our review of elementary statistical concepts that are commonly used in the financial industry (i.e. by prime brokerages, hedge funds, financial analysts, etc.). Recall from a recent article that the formula for a linear regression is:

Y_i = b₀ + b₁X_i + ε_i for i = 1, …, n

where:

Y_i is the i^th value of the dependent variable

b₀ is the y-intercept

b₁ is the slope coefficient

X_i is the i^th value of the independent variable

ε_i is the i^th value of an error term

i is the index of a particular variable

n is the maximum value of i

Unfortunately, we do not have access to the population values of b₀ and b₁, so we are forced to estimate these values with b_0estimated and b_1estimated.  This is one cause of uncertainty in the predicted value of Y_i. The second cause of uncertainty is the error term ε_i , which is the difference between the estimated and true value of Y_i.  These two uncertainties beg the question: How confident are we about the forecast results? To answer this question, we calculate a prediction interval which is an estimated interval into which future observations will fall, with a given probability, in light of past observations.

For example, if we forecasted that sales for ABC Corporation would grow by 8 percent this year, our prediction would be more meaningful if we were 95 percent confident that sales growth would fall in the interval from 7 percent to 9 percent.  A value outside the 7% to 9% range would not instill confidence in the value.

We can compute confidence intervals using our old friend, the standard error of the estimate s. The variance of the prediction error is equal to the square of the standard error of the estimate, namely s_f².   This estimated variance can be calculated using this formula:

Note that s_x² is the variance of the independent variable X.

After you calculate the variance of the prediction error, you choose a significance level α, say 0.05.  We apply another old friend, the t-statistic, which is the critical value for the forecast interval and can be looked up in the back of any statistics textbook..  By using (1 – α) = 0.95, we can compute the percent prediction interval Y as

Y = ± t_c s_f

Let’s take a numerical example[i] as follows:

1)     Assume a linear regression equation Y = 1.3478 + 30.0169(0.10) = 4.3495; the standard error of the estimate s = 0.7422; the mean value of X = 0.0647; the variance of the mean s_x² = 0.004641; the number of observations n = 9, the number of coefficients (the y-intercept and the slope) = 2.

2)     Assume we are interested in the 95% confidence interval.

3)     Compute the variance of the prediction error:

s_f² = 0.7422² [1 + 1/9 +  (0.10 – 0.0647)² /  (9 – 1)0.004641] = 0.630556

4)     Take the square root of the variance of the prediction error s_f², giving the standard deviation of the forecast error s_f = (0.630556)^1/2 =  0.7941.

5)     The degrees of freedom = (observations n – number of coefficients) = (9 – 2) = 7. From the back of a statistics book, the critical t-statistic for 7 degrees of freedom at the 95% confidence interval, t_c = 2.365.

6)     We compute the prediction interval for the 95% level of confidence. It is equal to the following:

Y = ± t_c s_f = 4.3495 – 2.365(0.7941) to 4.3495 + 2.365(0.7941) = 2.4715 to 6.2275.

From this example, we are 95% confident that a value of the dependent variable will have a value between 2.4715 and 6.2275, the prediction interval.

Well, we have now reviewed the basic concepts pertaining to single-variable linear regressions. We’ll pick up our voyage through financial statistics next time by examining multiple regressions.