Skip to content

CFA Level I · CFA Level I Exam

Applications of Simple Linear Regression in Finance for CFA Level I

Simple linear regression fits a straight line, Y = b0 + b1X + ε, that links one independent variable to one dependent variable. You estimate b0 and b1 by ordinary least squares, judge fit with R-squared and the standard error of estimate, test the slope with a t-statistic, then predict Y. Log forms handle non-linear relationships.

What this chapter covers

This chapter shows how to measure and test the relationship between two variables. A typical case is regressing a stock's excess return on the market's excess return, or regressing a company's revenue growth on GDP growth. You learn the model and its assumptions, how least squares picks the best-fit line, and how to judge whether that line is useful.

The chapter then moves to inference and use. You test whether the slope differs from zero, build a prediction interval around a forecast, and choose a functional form (log-lin, lin-log or log-log) when a straight line fits poorly. Most questions are short calculations that use a given table of regression output.

It draws on ideas from hypothesis testing, correlation and probability distributions. Beta in the CAPM and factor models use the same regression ideas. If you master this chapter, you read regression output quickly and avoid traps in other topics too.

Quantitative Methods carries 11-14% of the Level I exam, and this chapter is one of its most testable parts because the questions are compact and formula-driven. You get no penalty for wrong answers, but you cannot guess your way through numbers: with three options, a correct setup lets you eliminate two choices quickly. The same skills also help you with ANOVA tables, t-tests and beta in other topics. Most items need only a few lines of arithmetic, so this is a chapter where careful practice turns directly into reliable marks at about 90 seconds per question.

Applications of Simple Linear Regression in Finance: topics in the order to study them

  1. 1Simple Linear Regression Model and AssumptionsStart here: you need the model's terms (dependent, independent, intercept, slope, error) and its assumptions before any calculation makes sense.
  2. 2Ordinary Least Squares Estimation of Slope and InterceptNext, learn how the line is estimated, since slope = Cov(X,Y) ÷ Var(X) and the intercept feed every later topic.
  3. 3ANOVA, R-squared and Standard Error of EstimateOnce you have a fitted line, you measure how well it explains Y using sums of squares, R-squared and the standard error.
  4. 4Hypothesis Testing of Regression CoefficientsTesting the slope needs the standard error and degrees of freedom from the earlier topics, and it links to the F-test and correlation.
  5. 5Prediction Using Regression and Prediction IntervalsForecasting comes after you trust the model; the interval uses the standard error of the forecast and a critical t-value.
  6. 6Functional Forms: Log-Lin, Lin-Log and Log-Log ModelsStudy this last because it reuses everything above, only with transformed variables and a changed interpretation of the slope.

How to prepare Applications of Simple Linear Regression in Finance

Plan about five to seven short sessions. Keep the formulas on one page and practise reading regression output, because that is how the exam presents it.

  1. Write the model and the main assumptions in your own words: linear relationship, independent variable not random, constant error variance, uncorrelated errors, normally distributed errors.
  2. Compute a slope and intercept by hand for a small data set using b1 = Cov(X,Y) ÷ Var(X) and b0 = Ȳ − b1X̄. Then check with the TI BA II Plus: enter X and Y in the DATA worksheet (2nd DATA), then open the STAT worksheet (2nd STAT), set the method to LIN with 2nd SET, and scroll with the down arrow to read a, b and r.
  3. Learn the ANOVA table: SST = SSR + SSE, R² = SSR ÷ SST, standard error of estimate = √(SSE ÷ (n − 2)), and F = MSR ÷ MSE. Check each with a sample table.
  4. Practise t-tests: t = (b1 − hypothesised value) ÷ s(b1), with n − 2 degrees of freedom. Always state the decision rule before you calculate.
  5. Do prediction questions: compute Ŷ first, then add and subtract critical t × standard error of the forecast. Know that the interval widens as X moves away from its mean.
  6. For log forms, make a small table showing what each model means for the slope (a unit change or a percent change in X or Y). Then do timed sets of three-option questions and review every wrong answer.

Common mistakes in Applications of Simple Linear Regression in Finance

  • Using n − 1 degrees of freedom for the slope test

    Fix: In simple regression you estimate two parameters, so use n − 2 for the t-test and the standard error of estimate.

  • Treating a high R² as proof that the model is correct or that X causes Y

    Fix: R² only measures the share of variation in Y explained by X. Check the assumptions, the significance of the slope and the economic logic as well.

  • Mixing up SSR and SSE when computing R² or the standard error

    Fix: SSR is explained variation and goes in R² = SSR ÷ SST. SSE is unexplained variation and goes in the standard error √(SSE ÷ (n − 2)).

  • Forgetting to convert or interpret the log-form slope correctly

    Fix: First identify which variable is in logs. The slope's meaning changes between unit and percent changes, so write the interpretation before computing.

  • Using the standard error of the slope instead of the forecast standard error for a prediction interval

    Fix: A prediction interval for Y uses the standard error of the forecast. The slope's standard error is only for testing or building an interval for the slope.

  • Rounding too early in multi-step calculations

    Fix: Keep full calculator precision until the last step. Answer options can be close together, so early rounding may lead you to the wrong choice.

Last-day revision: Applications of Simple Linear Regression in Finance

  • Model: Y = b0 + b1X + ε, with one independent variable.
  • Slope b1 = Cov(X,Y) ÷ Var(X); intercept b0 = Ȳ − b1X̄.
  • The fitted line always passes through the point (X̄, Ȳ).
  • SST = SSR + SSE; R² = SSR ÷ SST.
  • In simple regression, R² equals the square of the correlation between X and Y.
  • Standard error of estimate = √(SSE ÷ (n − 2)); a lower value means a better fit.
  • Degrees of freedom for the slope t-test are n − 2.
  • Slope t-statistic = (b1 − hypothesised value) ÷ s(b1); reject H0 if |t| exceeds the critical value.
  • In simple regression, the F-statistic = MSR ÷ MSE equals t² for the slope.
  • A prediction interval is Ŷ ± tc × sf, and it is wider than the interval for the mean response.
  • Log-lin: ln Y on X; log-log: ln Y on ln X, where the slope is an elasticity-type measure.
  • Check residual plots for heteroskedasticity and non-normality before trusting the results.

Applications of Simple Linear Regression in Finance practice questions

Applications of Simple Linear Regression in Finance in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Applications of Simple Linear Regression in Finance: frequently asked questions

Do I need to memorise the t critical values for regression questions?

Usually the question gives the critical value or enough information to decide. It is still worth knowing that for large samples the 5% two-tailed value is close to 2. Focus more on the decision rule and the degrees of freedom.

Can I use the calculator to run a regression on the exam?

Yes. Both the approved TI BA II Plus and HP 12C can do linear regression, but the keystrokes differ. The TI uses the DATA and STAT worksheets. The HP 12C uses Σ+ to enter the data pairs and the ŷ,r function to get estimates and the correlation. Practise the one you will use, but many questions already give the output.

What is the difference between R² and the standard error of estimate?

R² is a proportion of variation in Y explained by the model, with no units. The standard error of estimate is the typical size of the residuals in the units of Y. You can use both to judge fit.

Why does the prediction interval get wider far from the mean of X?

The estimated line is least certain at extreme X values, so the forecast error grows as X moves away from the sample mean. The interval is narrowest at X̄.