Regression and hypothesis testing basics for FRM candidates
Regression estimates a relationship from data, such as how much a fund moves when the market moves. Hypothesis testing then asks whether the estimated relationship is distinguishable from noise. Both sit in the quantitative material of FRM® Part I. For the shape of the exam see how the FRM exam is structured, and for all topics see the FRM page.
The simple linear model
With one explanatory variable x and one outcome y, the model is:
y = a + b x + e
Here a is the intercept, b is the slope and e is an error term that captures everything else. Ordinary least squares (OLS) chooses a and b to minimise the sum of squared residuals. The hand formulas are:
- b = Sxy / Sxx, where Sxx is the sum of (x - mean x)^2 and Sxy is the sum of (x - mean x)(y - mean y).
- a = mean y - b x mean x.
The slope is the expected change in y for one unit of x. The intercept is the fitted y when x is zero.
Measuring fit and uncertainty
- Total variation Syy is the sum of (y - mean y)^2. The residual sum of squares, SSE, is the sum of squared residuals.
- R-squared = 1 - SSE / Syy: the share of the variation in y explained by the line. With one regressor it equals the squared correlation.
- Standard error of the regression: s = square root of SSE / (n - 2). The n - 2 reflects two estimated parameters.
- Standard error of the slope: se(b) = s / square root of Sxx.
Testing the slope
The usual null hypothesis is that the true slope is zero, meaning x tells you nothing about y. The test statistic is:
t = (b - 0) / se(b)
It follows a t distribution with n - 2 degrees of freedom if the errors are normal. The p-value is the probability, if the null were true, of a statistic at least as extreme as the one observed. A small p-value is evidence against the null, not the probability that it is true. A 95% confidence interval is b plus or minus t-critical x se(b), and it excludes zero exactly when the two-sided 5% test rejects.
Worked example
An invented fund, Pelham Strand Fund, has eight monthly observations. The market excess return x and the fund excess return y (both in per cent) are:
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| x | -2.0 | -1.0 | 0.5 | 1.0 | 1.5 | 2.5 | 3.0 | 4.0 |
| y | -1.4 | -0.2 | 0.9 | 1.1 | 2.4 | 2.6 | 3.9 | 4.4 |
Computed in Python from these formulas and checked against a library fit:
- Means: x = 1.1875, y = 1.7125. Sxx = 28.469, Sxy = 27.681, Syy = 27.649.
- Slope b = 27.681 / 28.469 = 0.9723. Intercept a = 1.7125 - 0.9723 x 1.1875 = 0.5578.
- SSE = 0.7332, so R-squared = 1 - 0.7332 / 27.649 = 0.9735.
- s = square root of (0.7332 / 6) = 0.3496. se(b) = 0.3496 / square root of 28.469 = 0.0655.
- t = 0.9723 / 0.0655 = 14.84 with 6 degrees of freedom. The two-sided p-value is about 0.000006.
- The t-critical value at 95% with 6 degrees of freedom is 2.447, so the interval is 0.9723 plus or minus 2.447 x 0.0655, or 0.812 to 1.133.
Reading it: the fund moves about 0.97% per 1% in the market, the slope is clearly nonzero, and 1 sits inside the interval, so a beta of 1 is not rejected at 5%. Eight points is far too few for real evidence; this shows the mechanics.
Usual mistakes
- Confusing correlation with causation. A significant slope shows association in this sample. It does not show that x causes y; a third variable or reverse causation can produce the same line.
- Overreading R-squared. A high value does not make the model correct or the forecasts reliable, and it can rise just by adding regressors. A low value does not make a significant slope useless.
- Using n - 1 instead of n - 2 degrees of freedom in the simple model.
- Treating "fail to reject" as proof of the null, or a p-value as the chance the result is a fluke.
- Ignoring assumptions (constant error variance, independent errors, linearity). Financial data often breaks them, as in fat tails, skewness and kurtosis.
How the exam typically tests it
Typically you are given regression output or summary sums and asked to compute a t-statistic, build a confidence interval, interpret R-squared, or decide whether a slope is significant. Charta's practice questions for this topic let you rehearse the arithmetic.
Common questions
What does R-squared tell me?
It is the fraction of the variation in the outcome that the fitted line accounts for in the sample. It says nothing on its own about causation, bias or out-of-sample accuracy.
How do I get the t-statistic for a slope?
Divide the estimated slope (less the hypothesised value, usually zero) by its standard error. Compare it with a t distribution with n - 2 degrees of freedom.
What does a p-value mean?
It is the probability of seeing a result at least as extreme as yours if the null hypothesis were true. It is not the probability that the null is true.
Why report a confidence interval?
It excludes zero exactly when the two-sided 5% test rejects, and it also shows the plausible size of the effect.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
The exam and the syllabus can change. Check GARP's current details before you book.