Linear regression models

 

Notes on linear regression analysis (pdf file)

Introduction to linear regression analysis

Mathematics of simple regression

Regression examples

·         Baseball batting averages

·         Beer sales vs. price, part 1: descriptive analysis

·         Beer sales vs. price, part 2: fitting a simple model

·         Beer sales vs. price, part 3: transformations of variables

·         Beer sales vs. price, part 4: additional predictors

·         NC natural gas consumption vs. temperature

What to look for in regression output

What’s a good value for R-squared?
What's the bottom line? How to compare models
Testing the assumptions of linear regression
Additional notes on regression analysis
Stepwise and all-possible-regressions
Excel file with simple regression formulas

Excel file with regression formulas in matrix form

If you are a PC Excel user, you must check this out:
RegressIt: free Excel add-in for linear regression and multivariate data analysis

 

Mathematics of simple regression


Review of the mean model

Formulas for the slope and intercept of a simple regression model

Formulas for R-squared and standard error of the regression

Formulas for standard errors and confidence limits for means and forecasts

Take-aways


Review of the mean model

 

To set the stage for discussing the formulas used to fit a simple (one-variable) regression model, let′s briefly review the formulas for the mean model, which can be considered as a constant-only (zero-variable) regression model.  You can use regression software to fit this model and produce all of the standard table and chart output by merely not selecting any independent variables.  R-squared will be zero in this case, because the mean model does not explain any of the variance in the dependent variable:  it merely measures it.

 

The forecasting equation of the mean model is:

...where b0 is the sample mean:

The sample mean has the (non-obvious) property that it is the value around which the mean squared deviation of the data is minimized, and the same least-squares criterion will be used later to estimate the "mean effect" of an independent variable.

 

The error that the mean model makes for observation t is therefore the deviation of Y from its historical average value:

The standard error of the model, denoted by s, is our estimate of the standard deviation of the noise in Y (the variation in it that is considered unexplainable). Smaller is better, other things being equal: we want the model to explain as much of the variation as possible. In the mean model, the standard error of the model is just is the sample standard deviation of Y:

 

(Here and elsewhere, STDEV.S denotes the sample standard deviation of X, using Excel notation. The population standard deviation is STDEV.P.) Note that the standard error of the model is not the square root of the average value of the squared errors within the historical sample of data. Rather, the sum of squared errors is divided by n-1 rather than n under the square root sign because this adjusts for the fact that a "degree of freedom for error″ has been used up by estimating one model parameter (namely the mean) from the sample of n data points.

 

The accuracy of the estimated mean is measured by the standard error of the mean, whose formula in the mean model is:

 

This is the estimated standard deviation of the error in estimating the mean. Notice that it is inversely proportional to the square root of the sample size, so it tends to go down as the sample size goes up. For example, if the sample size is increased by a factor of 4, the standard error of the mean goes down by a factor of 2, i.e., our estimate of the mean becomes twice as precise.

 

The accuracy of a forecast is measured by the standard error of the forecast, which (for both the mean model and a regression model) is the square root of the sum of squares of the standard error of the model and the standard error of the mean:

This is the estimated standard deviation of the error in the forecast, which is not quite the same thing as the standard deviation of the unpredictable variations in the data (which is s). It takes into account both the unpredictable variations in Y and the error in estimating the mean. In the mean model, the standard error of the mean is a constant, while in a regression model it depends on the value of the independent variable at which the forecast is computed, as explained in more detail below.

The standard error of the forecast gets smaller as the sample size is increased, but only up to a point. More data yields a systematic reduction in the standard error of the mean, but it does not yield a systematic reduction in the standard error of the model. The standard error of the model will change to some extent if a larger sample is taken, due to sampling variation, but it could equally well go up or down. The variations in the data that were previously considered to be inherently unexplainable remain inherently unexplainable if we continue to believe in the model′s assumptions, so the standard error of the model is always a lower bound on the standard error of the forecast.

 

Confidence intervals for the mean and for the forecast are equal to the point estimate plus-or-minus the appropriate standard error multiplied by the appropriate 2-tailed critical value of the t distribution. The critical value that should be used depends on the number of degrees of freedom for error (the number data points minus number of parameters estimated, which is n-1 for this model) and the desired level of confidence. It can be computed in Excel using the T.INV.2T function. So, for example, a 95% confidence interval for the forecast is given by

In general, T.INV.2T(0.05, n-1) is fairly close to 2 except for very small samples, i.e., a 95% confidence interval for the forecast is roughly equal to the forecast plus-or-minus two standard errors. (In older versions of Excel, this function was just called TINV.)   Return to top of page.

 


Formulas for the slope and intercept of a simple regression model:

 

Now let's regress. A simple regression model includes a single independent variable, denoted here by X, and its forecasting equation in real units is

It differs from the mean model merely by the addition of a multiple of Xt to the forecast. The estimated constant b0 is the Y-intercept of the regression line (usually just called "the intercept" or "the constant"), which is the value that would be predicted for Y at X = 0. The estimated coefficient b1 is the slope of the regression line, i.e., the predicted change in Y per unit of change in X. The simple regression model reduces to the mean model in the special case where the estimated slope is exactly zero. The estimated slope is almost never exactly zero (due to sampling variation), but if it is not significantly different from zero (as measured by its t-statistic), this suggests that the mean model should be preferred on grounds of simplicity unless there are good a priori reasons for believing that a relationship exists, even if it is largely obscured by noise.

Usually we do not care too much about the exact value of the intercept or whether it is significantly different from zero, unless we are really interested in what happens when X goes to "absolute zero" on whatever scale it is measured. Often X is a variable which logically can never go to zero, or even close to it, given the way it is defined. So, attention usually focuses mainly on the slope coefficient in the model, which measures the change in Y to be expected per unit of change in X as both variables move up or down relative to their historical mean values on their own natural scales of measurement.

 

The coefficients, standard errors, and forecasts for this model are obtained as follows.  First we need to compute the coefficient of correlation between Y and X, commonly denoted by rXY, which measures the strength of their linear relation on a relative scale of -1 to +1.  There are various formulas for it, but the one that is most intuitive is expressed in terms of the standardized values of the variables.  A variable is standardized by converting it to units of standard deviations from the mean.  The standardized version of X will be denoted here by X*, and its value in period t is defined in Excel notation as:

... where STDEV.P(X) is the population standard deviation, as noted above.  (Sometimes the sample standard deviation is used to standardize a variable, but the population standard deviation is needed in this particular formula.)   Y* will denote the similarly standardized value of Y. 

The correlation coefficient is equal to the average product of the standardized values of the two variables:

It is intuitively obvious that this statistic will be positive [negative] if X and Y tend to move in the same [opposite] direction relative to their respective means, because in this case X* and Y* will tend to have the same [opposite] sign.   Also, if X and Y are perfectly positively correlated, i.e., if Y is an exact positive linear function of X, then Y*t = X*t for all t, and the formula for rXY reduces to (STDEV.P(X)/STDEV.P(X))2, which is equal to 1. Similarly, an exact negative linear relationship yields rXY = -1.

 

The least-squares estimate of the slope coefficient (b1) is equal to the correlation times the ratio of the standard deviation of Y to the standard deviation of X:

The ratio of standard deviations on the RHS of this equation merely serves to scale the correlation coefficient appropriately for the real units in which the variables are measured. (The sample standard deviation could also be used here, because they only differ by a scale factor.)

 

The least-squares estimate of the intercept is the mean of Y minus the slope coefficient times the mean of X:

 

This equation implies that Y must be predicted to be equal to its own average value whenever X is equal to its own average value.

The standard error of the model (denoted again by s) is usually referred to as the standard error of the regression (or sometimes the "standard error of the estimate") in this context, and it is equal to the square root of {the sum of squared errors divided by n-2}, or equivalently, the standard deviation of the errors multiplied by the square root of (n-1)/(n-2), where the latter factor is a number slightly larger than 1:

The sum of squared errors is divided by n-2 in this calculation rather than n-1 because an additional degree of freedom for error has been used up by estimating two parameters (a slope and an intercept) rather than only one (the mean) in fitting the model to the data. The standard error of the regression is an unbiased estimate of the standard deviation of the noise in the data, i.e., the variations in Y that are not explained by the model.

Each of the two model parameters, the slope and intercept, has its own standard error, which is the estimated standard deviation of the error in estimating it. (In general, the term "standard error" means "standard deviation of the error" in whatever is being estimated. ) The standard error of the intercept is

which looks exactly like the formula for the standard error of the mean in the mean model, except for the additional term of (AVERAGE(X))2/VAR.P(X) under the square root sign. This term reflects the additional uncertainty about the value of the intercept that exists in situations where the center of mass of the independent variable is far from zero (in relative terms), in which case the intercept is determined by extrapolation far outside the data range. The standard error of the slope coefficient is given by:

...which also looks very similar, except for the factor of STDEV.P(X) in the denominator. Note that s is measured in units of Y and STDEV.P(X) is measured in units of X, so SEb1 is measured (necessarily) in "units of Y per unit of X", the same as b1 itself. The terms in these equations that involve the variance or standard deviation of X merely serve to scale the units of the coefficients and standard errors in an appropriate way.

You don′t need to memorize all these equations, but there is one important thing to note: the standard errors of the coefficients are directly proportional to the standard error of the regression and inversely proportional to the square root of the sample size. This means that noise in the data (whose intensity if measured by s) affects the errors in all the coefficient estimates in exactly the same way, and it also means that 4 times as much data will tend to reduce the standard errors of the all coefficients by approximately a factor of 2, assuming the data is really all generated from the same model, and a really huge of amount of data will reduce them to zero.

However, more data will not systematically reduce the standard error of the regression. As with the mean model, variations that were considered inherently unexplainable before are still not going to be explainable with more of the same kind of data under the same model assumptions. As the sample size gets larger, the standard error of the regression merely becomes a more accurate estimate of the standard deviation of the noise. Return to top of page.


Formulas for R-squared and standard error of the regression

 

The fraction of the variance of Y that is "explained" by the simple regression model, i.e., the percentage by which the sample variance of the errors ("residuals") is less than the sample variance of Y itself, is equal to the square of the correlation between them, i.e., "R squared":

Equivalently:

Thus, for example, if the correlation is rXY = 0.5, then rXY2 = 0.25, so the simple regression model explains 25% of the variance in Y in the sense that the sample variance of the errors of the simple regression model is 25% less than the sample variance of Y. This is not supposed to be obvious. It is a "strange but true" fact that can be proved with a little bit of calculus.

 

By taking square roots everywhere, the same equation can be rewritten in terms of standard deviations to show that the standard deviation of the errors is equal to the standard deviation of the dependent variable times the square root of 1-minus-the-correlation-squared:

However, the sample variance and standard deviation of the errors are not unbiased estimates of the variance and standard deviation of the unexplained variations in the data, because they do not into account the fact that 2 degrees of freedom for error have been used up in the process of estimating the slope and intercept. The fraction by which the square of the standard error of the regression is less than the sample variance of Y (which is the fractional reduction in unexplained variation compared to using the mean model) is the "adjusted" R-squared of the model, and in a simple regression model it is given by the formula

.

The factor of (n-1)/(n-2) in this equation is the same adjustment for degrees of freedom that is made in calculating the standard error of the regression. In fact, adjusted R-squared can be used to determine the standard error of the regression from the sample standard deviation of Y in exactly the same way that R-squared can be used to determine the sample standard deviation of the errors as a fraction of the sample standard deviation of Y:

 

 

You can apply this equation without even calculating the model coefficients or the actual errors!

 

In a multiple regression model with k independent variables plus an intercept, the number of degrees of freedom for error is n-(k+1), and the formulas for the standard error of the regression and adjusted R-squared remain the same except that the n-2 term is replaced by n-(k +1) .

 

It follows from the equation above that if you fit simple regression models to the same sample of the same dependent variable Y with different choices of X as the independent variable, then adjusted R-squared necessarily goes up as the standard error of the regression goes down, and vice versa. Hence, it is equivalent to say that your goal is to minimize the standard error of the regression or to maximize adjusted R-squared through your choice of X, other things being equal. However, as I will keep saying, the standard error of the regression is the real "bottom line" in your analysis: it measures the variations in the data that are not explained by the model in real economic or physical terms.

 

Adjusted R-squared can actually be negative if X has no measurable predictive value with respect to Y. In particular, if the correlation between X and Y is exactly zero, then R-squared is exactly equal to zero, and adjusted R-squared is equal to 1 - (n-1)/(n-2), which is negative because the ratio (n-1)/(n-2) is greater than 1. If this is the case, then the mean model is clearly a better choice than the regression model. Some regression software will not even display a negative value for adjusted R-squared and will just report it to be zero in that case. Return to top of page.

 


Formulas for standard errors and confidence limits for means and forecasts

 

The standard error of the mean of Y for a given value of X is the estimated standard deviation of the error in measuring the height of the regression line at that location, given by the formula

 

 

This looks like a lot like the formula for the standard error of the mean in the mean model: it is proportional to the standard error of the regression and inversely proportional to the square root of the sample size, so it gets steadily smaller as the sample size gets larger, approaching zero in the limit even in the presence of a lot of noise. However, in the regression model the standard error of the mean also depends to some extent on the value of X, so the term is scaled up by a factor that is greater than 1 and is larger for values of X that are farther from its mean, because there is relatively greater uncertainty about the true height of the regression line for values of X that are farther from its historical mean value.

 

 

The standard error for the forecast for Y for a given value of X is then computed in exactly the same way as it was for the mean model:

 

 

In the regression model it is larger for values of X that are farther from the mean--i.e., you expect to make bigger forecast errors when extrapolating the regression line farther out into space--because SEmean(X) is larger for more extreme values of X. The standard error of the forecast is not quite as sensitive to X in relative terms as is the standard error of the mean, because of the presence of the noise term s2 under the square root sign. (Remember that s2 is the estimated variance of the noise in the data.) In fact, s is usually much larger than SEmean(X) unless the data set is very small or X is very extreme, so usually the standard error of the forecast is not too much larger than the standard error of the regression.

 

 

Finally, confidence limits for means and forecasts are calculated in the usual way, namely as the forecast plus or minus the relevant standard error times the critical t-value for the desired level of confidence and the number of degrees of freedom, where the latter is n-2 for a simple regression model. For all but the smallest sample sizes, a 95% confidence interval is approximately equal to the point forecast plus-or-minus two standard errors, although there is nothing particularly magical about the 95% level of confidence. You can choose your own, or just report the standard error along with the point forecast.

 

 

Here are a couple of additional pictures that illustrate the behavior of the standard-error-of-the-mean and the standard-error-of-the-forecast in the special case of a simple regression model. Because the standard error of the mean gets larger for extreme (farther-from-the-mean) values of X, the confidence intervals for the mean (the height of the regression line) widen noticeably at either end.

 

The confidence intervals for predictions also get wider when X goes to extremes, but the effect is not quite as dramatic, because the standard error of the regression (which is usually a bigger component of forecast error) is a constant. Note that the inner set of confidence bands widens more in relative terms at the far left and far right than does the outer set of confidence bands.

 

But remember: the standard errors and confidence bands that are calculated by the regression formulas are all based on the assumption that the model is correct, i.e., that the data really is described by the assumed linear equation with normally distributed errors. If the model assumptions are not correct--e.g., if the wrong variables have been included or important variables have been omitted or if there are non-normalities in the errors or nonlinear relationships among the variables--then the predictions and their standard errors and confidence limits may all be suspect. So, when we fit regression models, we don′t just look at the printout of the model coefficients. We look at various other statistics and charts that shed light on the validity of the model assumptions. Return to top of page.

 


 

Take-aways

 

1. The coefficients and error measures for a regression model are entirely determined by the following summary statistics: means, standard deviations and correlations among the variables, and the sample size.

 

2. The correlation between Y and X , denoted by rXY, is equal to the average product of their standardized values, i.e., the average of {the number of standard deviations by which Y deviates from its mean} times {the number of standard deviations by which X deviates from its mean}, using the population (rather than sample) standard deviation in the calculation.  This statistic measures the strength of the linear relation between Y and X on a relative scale of -1 to +1.  The correlation between Y and X is positive if they tend to move in the same direction relative to their respective means and negative if they tend to move in opposite directions, and it is zero if their up-or-down movements with respect to their own means are statistically independent.

 

3. The slope coefficient in a simple regression of Y on X is the correlation between Y and X multiplied by the ratio of their standard deviations:

 

Either the population or sample standard deviation (STDEV.S) can be used in this formula because they differ only by a multiplicative factor.

4. In a simple regression model, the percentage of variance "explained" by the model, which is called R-squared, is the square of the correlation between Y and X. That is, R-squared = rXY2, and that′s why it′s called R-squared. This means that the sample standard deviation of the errors is equal to {the square root of 1-minus-R-squared} times the sample standard deviation of Y:

STDEV.S(errors) = (SQRT(1 minus R-squared)) x STDEV.S(Y).

So, if you know the standard deviation of Y, and you know the correlation between Y and X, you can figure out what the standard deviation of the errors would be be if you regressed Y on X. However...

 

5. The sample standard deviation of the errors is a downward-biased estimate of the size of the true unexplained deviations in Y because it does not adjust for the additional "degree of freedom" used up by estimating the slope coefficient. An unbiased estimate of the standard deviation of the true errors is given by the standard error of the regression, denoted by s. In the special case of a simple regression model, it is:

Standard error of regression = STDEV.S(errors) x SQRT((n-1)/(n-2))

This is the real bottom line, because the standard deviations of the errors of all the forecasts and coefficient estimates are directly proportional to it (if the model′s assumptions are correct!!)

6. Adjusted R-squared, which is obtained by adjusting R-squared for the degrees if freedom for error in exactly the same way, is an unbiased estimate of the amount of variance explained:

Adjusted R-squared = 1 - ((n-1)/(n-2)) x (1 - R-squared).

For large values of n, there isn′t much difference.

 

In a multiple regression model in which k is the number of independent variables, the n-2 term that appears in the formulas for the standard error of the regression and adjusted R-squared merely becomes n-(k+1).

 

7. The important thing about adjusted R-squared is that:

Standard error of the regression = (SQRT(1 minus adjusted-R-squared)) x STDEV.S(Y).

So, for models fitted to the same sample of the same dependent variable, adjusted R-squared always goes up when the standard error of the regression goes down.

 

A model does not always improve when more variables are added: adjusted R-squared can go down (even go negative) if irrelevant variables are added.

 

8. The standard error of a coefficient estimate is the estimated standard deviation of the error in measuring it. Also, the estimated height of the regression line for a given value of X has its own standard error, which is called the standard error of the mean at X. All of these standard errors are proportional to the standard error of the regression divided by the square root of the sample size. So a greater amount of "noise" in the data (as measured by s) makes all the estimates of means and coefficients proportionally less accurate, and a larger sample size makes all of them more accurate (4 times as much data reduces all the standard errors by a factor of 2, etc.). However, more data will not systematically reduce the standard error of the regression. Rather, the standard error of the regression will merely become a more accurate estimate of the true standard deviation of the noise.

 

9. The standard error of the forecast for Y at a given value of X is the square root of the sum of squares of the standard error of the regression and the standard error of the mean at X. The standard error of the mean is usually a lot smaller than the standard error of the regression except when the sample size is very small and/or you are trying to predict what will happen under very extreme conditions (which is dangerous), so the standard error of the forecast is usually only slightly larger than the standard error of the regression. (Recall that under the mean model, the standard error of the mean is a constant. In a simple regression model, the standard error of the mean depends on the value of X, and it is larger for values of X that are farther from its own mean.)

 

10. Two-sided confidence limits for coefficient estimates, means, and forecasts are all equal to their point estimates plus-or-minus the appropriate critical t-value times their respective standard errors. For a simple regression model, in which two degrees of freedom are used up in estimating both the intercept and the slope coefficient, the appropriate critical t-value is T.INV.2T(1 - C, n - 2) in Excel, where C is the desired level of confidence and n is the sample size. The usual default value for the confidence level is 95%, for which the critical t-value is T.INV.2T(0.05, n - 2).

 

The accompanying  Excel file with simple regression formulas shows how the calculations described above can be done on a spreadsheet, including a comparison with output from RegressIt.  For the case in which there are two or more independent variables, a so-called multiple regression model, the calculations are not too much harder if you are familiar with how to do arithmetic with vectors and matrices. Here is an Excel file with regression formulas in matrix form that illustrates this process.    Return to top of page.

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