III Causal Forecasting Methods

These methods use historical data to estimate the relationship between the variable you are trying to predict (a.k.a. dependent variable or response variable) and other variables (a.k.a. independent variables or explanatory variables).

For example, we might try to predict a driver’s reaction time using the number of hours of sleep the driver had the night before and the number of ounces of alcohol he/she has consumed. In this case, reaction time is the dependent variable and sleep hours and alcohol ounces are the independent variables.

Another example might involve estimating someone’s income (the dependent variable) based on his/her age and number of years of higher education (the independent variables).

The forecasting model will always involve only one dependent variable and one or more independent variables.

IIIA Simple Linear Regression

This model is appropriate when there is only one independent variable.

The population model is stated as

y_i = b ₀ + b ₁ x _i + e _I

where

y_i = observed value of the dependent variable for case i

b ₀ = the population intercept

b ₁ = the population slope

x _i = the observed value of the independent variable

e _i = the error for case i (can be positive or negative)

This model is fitted using sample data and the sample model can be stated as

yhat _i = b ₀ + b ₁ x _i

where

yhat _i = expected or predicted value of the dependent variable for case i

b ₀ = the intercept (if x equals zero, the expected value of y is b ₀ )

b ₁ = the slope (for every one unit increase in x, y is expected to change by b ₁ units)

x _i = the value of the independent variable for case i

Let e _i = y _i – yhat _i be the error for case i

But how do you come up with b ₀ and b ₁? They are determined by minimizing the sum of the squared error terms. This is known as least squares estimation.

b ₁ = [S (x_i – xbar)(y_i-ybar)]/S(_xi – xbar)²

b ₀ = ybar - b ₁xbar

where ybar is the average value of the dependent variable and xbar is the average value of the independent variable.

To illustrate, look at SODA.xls on the formulas page.

The resulting model is

yhat _i = 10.90 + 0.90 x _i

So if attendance is 200 (first verify that it is in the range for the sample). We can forecast soda consumption by calculating

yhat _i = 10.90 + 0.90 * 200 = 190.90 sodas

For each additional person attending the game, soda consumption is expected to increase by 0.90 sodas.

If no people went to the game, expected consumption is 10.90 sodas (??)

The intercept does not always have a good intuitive meaning.

We can achieve the same results as well as some additional analysis using some built-in functions of Excel. Look at SODA.XLS on the built-in page.

First name ranges. Highlight cells b6:b25 and name this block ATTEND. Highlight c6:c25 and name this block SODAS.

In cell b28 type =intercept(sodas,attend)

In cell b29 type =slope(sodas,attend)

In cell b30 type =rsq(sodas,attend)

In cell b31 type =steyx(sodas,attend)

In cell b32 type =correl(sodas,attend)

We already know what the intercept and slope represent, but what about the others? First some background…

SST = SSR + SSE

Sum of Squares Total (SST) is the total variability of the y values around their mean.

Sum of Squares Regression (SSR) is the amount of variability explained by the regression relationship.

Sum of Squares Error (SSE) is the amount of variability not explained (recall that we minimize the sum of the squared error terms).

Want SSE è 0 and SSR è SST

Coefficient of Determination R² = SSR/SST Proportion of variability in Y explained by x. [rsq(y range, x range)]

0<= R² <= 1

If R² = 1, perfect relationship

If R² = 0, no relationship

For the soda problem R² = 0.96 which means that 96% of the variability in soda consumption can be explained by attendance.

Related to that is the Correlation Coefficient r_xy = sqrt(R²) [=correl(y range, x range)]

It takes the sign of the slope.

-1 <= r_xy <= 1

If r_xy = 1, perfect positive correlation

If r_xy = -1, perfect negative correlation

If r_xy = 0, no correlation

Standard Error of the Estimate s_e is a measure of forecasting accuracy. [=steyx(yrange, x range)]

68% of actual values will be within +/- one s_e of the forecasted value.

95% of actual values will be within +/- two s_e of the forecasted value.

Can construct a scatter plot of these observations.

Highlight ATTEND and SODAS.

Click on chart wizard.

XY scatter

Next

Next

Titles Soda consumption versus Attendance

Next

As new sheet

Finish

Now right mouse click on any point.

Choose Add Trendline

Linear

OK

Can also run a regression using Excel’s Data Analysis tools.

Tools

Data Analysis (if this option does not appear, you need to go to add-ins and select it)

Regression

OK

Input Y Range c5:c25

Input X Range b5:b25

Check labels box

Output Range any blank space

Check Residuals box

Check Residual Plots box

OK