Posts Tagged ‘dependent variable’

Forecast Friday Topic: Selecting the Variables for a Regression

October 14, 2010

(Twenty-fifth in a series)

When it comes to building a regression model, for many companies there’s good news and bad news. The good news: there’s plenty of independent variables from which to choose. The bad news: there’s plenty of independent variables from which to choose! While it may be possible to run a regression with all possible independent variables, each one included in your model reduces your degrees of freedom and causes the model to overfit the data on which the model is built, resulting in less reliable forecasts when new data is introduced.

So how do you come up with your short list of independent variables?

Some analysts have tried plotting the dependent variable (Y) against individual independent variables (Xi) and selecting it if there’s some noticeable relationship. Another tried method is to produce a correlation matrix of all the independent variables and if a large correlation between two of them is discovered, drop one from consideration (so to avoid multicollinearity). Still another approach has been to perform a multiple linear regression on all possible explanatory variables and then dropping those who t values are insignificant. These approaches are often selected because they are quick and simple, but they are not reliable for coming up with a decent regression model.

Stepwise Regression

Other approaches are a bit more complex, but more reliable. Perhaps the most common of these approaches is stepwise regression. Stepwise regression works by first identifying the independent variable with the highest correlation with the dependent variable. Once that variable is identified, a one-variable regression model is run. The residuals of that model are then obtained. Recall from previous Forecast Friday posts that if an important variable is omitted from a regression model, its effect on the dependent variable gets factored into the residuals. Hence, the next step in a stepwise regression is to identify the one unselected independent variable with the highest correlation with the residuals. Now you have your second independent variable, and you run a two-variable regression model. You then look at the residuals to that model and select the independent variable with the highest correlation to them, and so forth. Repeat the process until no more variables can be added into the model.

Many statistical analysis packages do stepwise regression seamlessly. Stepwise regression is not guaranteed to produce the optimal set of variables for your model.

Other Approaches

Other approaches to variable selection include best subsets regression, which involves taking various subsets of the available independent variables and running models with them, choosing the subset with the best R2. Many statistical software packages have the capability of helping determine the various subsets to choose from. Principal components analysis of all the variables is another approach, but it is beyond the scope of this discussion.

Despite systematic techniques like stepwise regression, variable selection in regression models is as much an art as a science. Whatever variables you select for your model should have a valid rationale for being there.

Next Forecast Friday Topic: I haven’t decided yet!

Let me surprise you. In the meantime, have a great weekend and be well!

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Forecast Friday Topic: Heteroscedasticity

August 12, 2010

(Seventeenth in a series)

Recall that one of the important assumptions in regression analysis is that a regression equation exhibit homoscedasticity: the condition that the error terms have a constant variance. Today we discuss heteroscedasticity, the violation of that assumption.

Heteroscedasticity, like autocorrelation and multicollinearity, results in inefficient parameter estimates. The standard errors of the parameter estimates tend to be biased, which means that the t-ratios and confidence intervals calculated around the suspect independent variable will not be valid, and will generate dubious predictions.

Heteroscedasticity occurs mostly in cross-sectional, as opposed to time series, data and mostly in large data sets. When data sets are large, the range of values for an independent variable can be quite wide. This is especially the case in data where income or other measures of wealth are used as independent variables. Persons with low income have few options about how to spend their money while persons with high incomes have many options. If you were trying to predict that the conviction rate for crimes was different in low income counties vs. high income counties, your model may exhibit heteroscedasticity because a low-income person may not have the funds for an adequate defense, and may be restricted to a public defender, or other inexpensive attorney. A wealthy individual, on the other hand, can hire the very best defense lawyer money could buy; or he could choose an inexpensive lawyer, or even the public defender. The wealthy individual may even be able to make restitution in lieu of a conviction.

How does this disparity affect your model? Recall from our earlier discussions on regression analysis that the least-squares method places more weight on extreme values. When outliers exist in data, they generate large residuals that get scattered out from those of the remaining observations. While heteroscedastic error terms will still have a mean of zero, their variance is greatly out of whack, resulting in inefficient parameter estimates.

In today’s Forecast Friday post, we will look at a data set for a regional housing market, perform a regression, and show how to detect heteroscedasticity visually.

Heteroscedasticity in the Housing Market

The best depiction of heteroscedasticity comes from my college econometrics textbook, Introducing Econometrics, by William S. Brown. In the chapter on heteroscedasticity, Brown provides a data set of housing statistics from the 1980 Census for Pierce County, Washington, which I am going to use for our model. The housing market is certainly one market where heteroscedasticity is deeply entrenched, since there is a dramatic range for both incomes and home market values. In our data set, we have 59 census tracts within Pierce County. Our independent variable is the median family income for the census tract; our dependent variable is the OwnRatio – the ratio of the number of families who own their homes to the number of families who rent. Our data set is as follows:

Housing Data

Tract

Income

Ownratio

601

$24,909

7.220

602

$11,875

1.094

603

$19,308

3.587

604

$20,375

5.279

605

$20,132

3.508

606

$15,351

0.789

607

$14,821

1.837

608

$18,816

5.150

609

$19,179

2.201

609

$21,434

1.932

610

$15,075

0.919

611

$15,634

1.898

612

$12,307

1.584

613

$10,063

0.901

614

$5,090

0.128

615

$8,110

0.059

616

$4,399

0.022

616

$5,411

0.172

617

$9,541

0.916

618

$13,095

1.265

619

$11,638

1.019

620

$12,711

1.698

621

$12,839

2.188

623

$15,202

2.850

624

$15,932

3.049

625

$14,178

2.307

626

$12,244

0.873

627

$10,391

0.410

628

$13,934

1.151

629

$14,201

1.274

630

$15,784

1.751

631

$18,917

5.074

632

$17,431

4.272

633

$17,044

3.868

634

$14,870

2.009

635

$19,384

2.256

701

$18,250

2.471

705

$14,212

3.019

706

$15,817

2.154

710

$21,911

5.190

711

$19,282

4.579

712

$21,795

3.717

713

$22,904

3.720

713

$22,507

6.127

714

$19,592

4.468

714

$16,900

2.110

718

$12,818

0.782

718

$9,849

0.259

719

$16,931

1.233

719

$23,545

3.288

720

$9,198

0.235

721

$22,190

1.406

721

$19,646

2.206

724

$24,750

5.650

726

$18,140

5.078

728

$21,250

1.433

731

$22,231

7.452

731

$19,788

5.738

735

$13,269

1.364

Data taken from U.S. Bureau of Census 1980 Pierce County, WA; Reprinted in Brown, W.S., Introducing Econometrics, St. Paul (1991): 198-200.

When we run our regression, we get the following equation:

Ŷ= 0.000297*Income – 2.221

Both the intercept and independent variable’s parameter estimates are significant, with the intercept parameter having a t-ratio of -4.094 and the income estimate having one of 9.182. R2 is 0.597, and the F-statistic is a strong 84.31. The model seems to be pretty good – strong t-ratios and F-statistic, a high coefficient of determination, and the sign on the parameter estimate for Income is positive, as we would expect. Generally, the higher the income, the greater the Own-to-rent ratio. So far so good.

The problem comes when we do a visual inspection of our data: first the independent variable against the dependent variable and the independent variable against the regression residuals. First, let’s take a look at the scatter plot of Income and OwnRatio:

Without even looking at the residuals, we can see that as median family income increases, the data points begin to spread out. Look at what happens to the distance between data points above and below the line when median family incomes reach $20,000: OwnRatios vary drastically.

Now let’s plot Income against the regression’s residuals:

This scatter plot shows essentially the same phenomenon as the previous graph, but from a different perspective. We can clearly see the error terms fanning out as Income increases. In fact, we can see the residuals diverging at increasing rates once Income starts moving from $10,000 to $15,000, and just compounding as incomes go higher. Roughly half the residuals fall on both the positive and the negative side, allowing us to meet the regression assumption of our residuals having a mean of zero, hence our parameter estimates are not biased. However, because we violated the constant variance assumption, the standard error of our regression is biased, so our parameter estimates are suspect.

Visual Inspection Only Gets You So Far

By visually inspecting our residuals, we can clearly see that our error terms are not homoscedastic. When you have a regression model, especially for cross-sectional data sets like this, you should visually inspect every independent variable against the dependent variable and against the error terms in order to get a priori indication of heteroscedasticity. However, visual inspection alone is not a guarantee that heteroscedasticity exists. There are three particularly simple methods to detecting heteroscedasticity which we will discuss in next week’s Forecast Friday post: the Park Test, the Goldfeld-Quandt Test, and the Breusch-Pagan Test.

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Forecast Friday Topic: Multicollinearity – Correcting and Accepting it

July 22, 2010

(Fourteenth in a series)

In last week’s Forecast Friday post, we discussed how to detect multicollinearity in a regression model and how dropping a suspect variable or variables from the model can be one approach to reducing or eliminating multicollinearity. However, removing variables can cause other problems – particularly specification bias – if the suspect variable is indeed an important predictor. Today we will discuss two additional approaches to correcting multicollinearity – obtaining more data and transforming variables – and will discuss when it’s best to just accept the multicollinearity.

Obtaining More Data

Multicollinearity is really an issue with the sample, not the population. Sometimes, sampling produces a data set that might be too homogeneous. One way to remedy this would be to add more observations to the data set. Enlarging the sample will introduce more variation in the data series, which reduces the effect of sampling error and helps increase precision when estimating various properties of the data. Increased sample sizes can reduce either the presence or the impact of multicollinearity, or both. Obtaining more data is often the best way to remedy multicollinearity.

Obtaining more data does have problems, however. Sometimes, additional data just isn’t available. This is especially the case with time series data, which can be limited or otherwise finite. If you need to obtain that additional information through great effort, it can be costly and time consuming. Also, the additional data you add to your sample could be quite similar to your original data set, so there would be no benefit to enlarging your data set. The new data could even make problems worse!

Transforming Variables

Another way statisticians and modelers go about eliminating multicollinearity is through data transformation. This can be done in a number of ways.

Combine Some Variables

The most obvious way would be to find a way to combine some of the variables. After all, multicollinearity suggests that two or more independent variables are strongly correlated. Perhaps you can multiply two variables together and use the product of those two variables in place of them.

So, in our example of the donor history, we had the two variables “Average Contribution in Last 12 Months” and “Times Donated in Last 12 Months.” We can multiply them to create a composite variable, “Total Contributions in Last 12 Months,” and then use that new variable, along with the variable “Months Since Last Donation” to perform the regression. In fact, if we did that with our model, we end up with a model (not shown here) that has an R2=0.895, and this time the coefficient for “Months Since Last Donation” is significant, as is our “Total Contribution” variable. Our F statistic is a little over 72. Essentially, the R2 and F statistics are only slightly lower than in our original model, suggesting that the transformation was useful. However, looking at the correlation matrix, we still see a strong negative correlation between our two independent variables, suggesting that we still haven’t eliminated multicollinearity.

Centered Interaction Terms

Sometimes we can reduce multicollinearity by creating an interaction term between variables in question. In a model trying to predict performance on a test based on hours spent studying and hours of sleep, you might find that hours spent studying appears to be related with hours of sleep. So, you create a third independent variable, Sleep_Study_Interaction. You do this by computing the average value for both the hours of sleep and hours of studying variables. For each observation, you subtract each independent variable’s mean from its respective value for that observation. Once you’ve done that for each observation, multiply their differences together. This is your interaction term, Sleep_Study_Interaction. Run the regression now with the original two variables and the interaction term. When you subtract the means from the variables in question, you are in effect centering interaction term, which means you’re taking into account central tendency in your data.

Differencing Data

If you’re working with time series data, one way to reduce multicollinearity is to run your regression using differences. To do this, you take every variable – dependent and independent – and, beginning with the second observation – subtract the immediate prior observation’s values for those variables from the current observation. Now, instead of working with original data, you are working with the change in data from one period to the next. Differencing eliminates multicollinearity by removing the trend component of the time series. If all independent variables had followed more or less the same trend, they could end up highly correlated. Sometimes, however, trends can build on themselves for several periods, so multiple differencing may be required. In this case, subtracting the period before was taking a “first difference.” If we subtracted two periods before, it’s a “second difference,” and so on. Note also that with differencing, we lose the first observations in the data, depending on how many periods we have to difference, so if you have a small data set, differencing can reduce your degrees of freedom and increase your risk of making a Type I Error: concluding that an independent variable is not statistically significant when, in truth it is.

Other Transformations

Sometimes, it makes sense to take a look at a scatter plot of each independent variable’s values with that of the dependent variable to see if the relationship is fairly linear. If it is not, that’s a cue to transform an independent variable. If an independent variable appears to have a logarithmic relationship, you might substitute its natural log. Also, depending on the relationship, you can use other transformations: square root, square, negative reciprocal, etc.

Another consideration: if you’re predicting the impact of violent crime on a city’s median family income, instead of using the number of violent crimes committed in the city, you might instead divide it by the city’s population and come up with a per-capita figure. That will give more useful insights into the incidence of crime in the city.

Transforming data in these ways helps reduce multicollinearity by representing independent variables differently, so that they are less correlated with other independent variables.

Limits of Data Transformation

Transforming data has its own pitfalls. First, transforming data also transforms the model. A model that uses a per-capita crime figure for an independent variable has a very different interpretation than one using an aggregate crime figure. Also, interpretations of models and their results get more complicated as data is transformed. Ideally, models are supposed to be parsimonious – that is, they explain a great deal about the relationship as simply as possible. Typically, parsimony means as few independent variables as possible, but it also means as few transformations as possible. You also need to do more work. If you try to plug in new data to your resulting model for forecasting, you must remember to take the values for your data and transform them accordingly.

Living With Multicollinearity

Multicollinearity is par for the course when a model consists of two or more independent variables, so often the question isn’t whether multicollinearity exists, but rather how severe it is. Multicollinearity doesn’t bias your parameter estimates, but it inflates their variance, making them inefficient or untrustworthy. As you have seen from the remedies offered in this post, the cures can be worse than the disease. Correcting multicollinearity can also be an iterative process; the benefit of reducing multicollinearity may not justify the time and resources required to do so. Sometimes, any effort to reduce multicollinearity is futile. Generally, for the purposes of forecasting, it might be perfectly OK to disregard the multicollinearity. If, however, you’re using regression analysis to explain relationships, then you must try to reduce the multicollinearity.

A good approach is to run a couple of different models, some using variations of the remedies we’ve discussed here, and comparing their degree of multicollinearity with that of the original model. It is also important to compare the forecast accuracy of each. After all, if all you’re trying to do is forecast, then a model with slightly less multicollinearity but a higher degree of forecast error is probably not preferable to a more precise forecasting model with higher degrees of multicollinearity.

The Takeaways:

  1. Where you have multiple regression, you almost always have multicollinearity, especially in time series data.
  2. A correlation matrix is a good way to detect multicollinearity. Multicollinearity can be very serious if the correlation matrix shows that some of the independent variables are more highly correlated with each other than they are with the dependent variable.
  3. You should suspect multicollinearity if:
    1. You have a high R2 but low t-statistics;
    2. The sign for a coefficient is opposite of what is normally expected (a relationship that should be positive is negative, and vice-versa).
  4. Multicollinearity doesn’t bias parameter estimates, but makes them untrustworthy by enlarging their variance.
  5. There are several ways of remedying multicollinearity, with obtaining more data often being the best approach. Each remedy for multicollinearity contributes a new set of problems and limitations, so you must weigh the benefit of reduced multicollinearity on time and resources needed to do so, and the resulting impact on your forecast accuracy.

Next Forecast Friday Topic: Autocorrelation

These past two weeks, we discussed the problem of multicollinearity. Next week, we will discuss the problem of autocorrelation – the phenomenon that occurs when we violate the assumption that the error terms are not correlated with each other. We will discuss how to detect autocorrelation, discuss in greater depth the Durbin-Watson statistic’s use as a measure of the presence of autocorrelation, and how to correct for autocorrelation.

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Forecast Friday Topic: Multicollinearity – How to Detect it; How to Correct it

July 15, 2010

(Thirteenth in a series)

In last week’s Forecast Friday post, we explored how to perform regression analysis using Excel. We looked at the giving history of 20 contributors to a nonprofit organization, and developed a model based on the recency, frequency, and monetary value (RFM) of their past donations. We derived the following regression equation:

We were pleased to see that our model had a coefficient of determination – or R2=0.933, indicating that our model explained 93.3% of the change in the donor’s current contribution (our Ŷ). But we were a little disheartened when we looked at the t-statistics of each of our regression coefficients. Recall that we found our recency coefficient was not significant:

Parameter

Coefficient

T-statistic

Significant?

Intercept

87.27

4.32

Yes

Months since Last

(1.80)

(1.44)

No

Times Donated

2.45

2.87

Yes

Average Contribution

0.35

3.26

Yes

Yet, most direct marketing professionals know clearly that RFM theory postulates that all three variables are significant indicators of whether and how much a donor will give (or a customer will buy). When our model doesn’t replicate what a tried and true theory has long maintained, there could possibly be something wrong.

Multicollinearity

Most times, when something doesn’t look right in the results of a regression model, it is safe to assume that one of the regression assumptions has been violated. The problem is trying to determine which assumption – or assumptions – was violated. Since the coefficient for “Months Since Last Contribution” has a t-statistic that indicates it isn’t statistically significant, we might suspect that the specification assumption is violated: that is, we may believe that “Months Since Last Contribution” is an extraneous, irrelevant variable that should not have been included in the model and, thus, be removed.

But is that really the case? There can be other reasons why a parameter estimate does not come up significant. If two or more independent variables are highly correlated, the resulting multicollinearity can cause the regression model to assign a statistically insignificant parameter estimate to an important independent variable. So, how can we detect multicollinearity?

Detecting Multicollinearity: Correlation Matrix

The first step in detecting multicollinearity is to examine the correlation among the independent variables. We do this by looking at a correlation matrix. You can run a correlation matrix in Excel by using its Data Analysis ToolPak. Looking at the correlation matrix for our variables, we find:

Correlation Matrix – Original Variables

Variable

Contribution Y

Months Since Last Donation X1

Times Donated in last 12 months

X2

Average Contribution in last 12 months

X3

Contribution (Y)

1.00

  

  

  

Months Since Last Donation – X1

-0.93

1.00

  

  

Times Donated in last 12 months – X2

0.89

-0.88

1.00

  

Average Contribution Last 12 mo. – X3

0.88

-0.84

0.69

1.00

 

A correlation of 1.00 means two variables are perfectly correlated; a correlation of 0.00 means there is absolutely no correlation. The cells in the matrix above, where the correlation is 1.00, shows the correlation of an independent variable with itself – we would expect a perfectly correlated relationship. What is most important to us are the numbers below the 1.00 correlations. The first column shows our dependent variable, “Contribution”. As you go down the column, row by row, you see that each of our independent variables is strongly correlated with the dependent variable, indicating that they are all strong predictors.

The correlation between “Months Since Last Donation” (X1) and the donor’s Contribution (Y) shows a correlation that is almost perfectly negative (-0.93), while those correlations of the dependent variable with each of the other two independent variables is almost perfectly positive with the contribution (0.89 and 0.88). When writing these in shorthand, we use the Greek letter rho, ρ, to denote correlation. Hence, to show the correlation between each independent variable with the dependent variable, we would express them as follows:

ρX1Y = -0.93

ρX2Y = 0.89

ρX3Y = 0.88

But now, let’s look at the correlations among our independent variables:

ρX1X2= -0.88

ρX1X3= -0.84

ρX2X3= 0.69

 

Notice that all of our independent variables are highly correlated with one another. The relationship between “Times Donated in Last 12 Months” and “Average Contribution in Last 12 Months” is not as strong as the correlation between those individual variables with “Months Since Last Donation,” but the correlation is still very strong.

Hence, we can conclude that multicollinearity is present in this model.

Correcting Multicollinearity: Dropping Variables

In today’s post, we will discuss one of the remedies for multicollinearity – dropping a highly correlated independent variable. Next week, we’ll discuss the other approaches to correcting multicollinearity. Sometimes, when a variable is “iffy,” we can save ourselves some trouble and just kick it out. If we were to ignore “Months Since Last Donation,” and run our regression with the remaining two variables, we end up with the following regression equation:

Ŷ= 60.68 + 3.37X2 + 0.45X3

We get R2 =0.924, suggesting that we didn’t lose much explanatory power by excluding “Months Since Last Donation.” We also get an F statistic of 103.36, much higher than the 73.90 we had in our original model. A higher F-statistic indicates a model that is more statistically valid. It also reflects the exclusion of one or more extraneous variables. Also, the t-statistics for both independent variables are significant, and they’re even higher than they were in the original model, further indicating increased validity:

Parameter

Coefficient

T-statistic

Significant?

Intercept

60.68

7.24

Yes

Times Donated

3.37

5.83

Yes

Average Contribution

0.45

5.49

Yes

Dropping “Months Since Last Donation” from our analysis worked here. However, dropping variables without a rational decision process can cause new problems. In some cases, dropping a variable can result in specification bias, as we saw in our previous example of predicting profit margin for savings and loan associations a few weeks ago. So, consider dropping variables cautiously.

Next Forecast Friday Topic: More Multicollinearity Remedies

Today, we described one of the ways to remedy multicollinearity – dropping variables. Next week, we will explore two other ways of correcting multicollinearity: obtaining more data and transforming variables. We will also discuss the pitfalls of all three of these remedies, and we will discuss when it’s not worth it to reduce the impact of multicollinearity.

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Forecast Friday Topic: Multiple Regression Analysis (continued)

June 24, 2010

(Tenth in a series)

Today we resume our discussion of multiple regression analysis. Last week, we built a model to determine the extent of any relationship between U.S. savings & loan associations’ percent profit margin and two independent variables, net revenues per deposit dollar and number of S&L offices. Today, we will compute the 95% confidence interval for each parameter estimate; determine whether the model is valid; check for autocorrelation; and use the model to forecast. Recall that our resulting model was:

Yt = 1.56450 + 0.23720X1t – 0.000249X2t

Where Yt is the percent profit margin for the S&L in Year t; X1t is the net revenues per deposit dollar in Year t; and X2t is the number of S&L offices in the U.S. in Year t. Recall that the R2 is .865, indicating that 86.5% of the change in percentage profit margin is explained by changes in net revenues per deposit dollar and number of S&L offices.

Determining the 95% Confidence Interval for the Partial Slope Coefficients

In multiple regression analysis, since there are multiple independent variables, the parameter estimates for each independent variable both impact the slope of the line; hence the coefficients β1t and β2t are referred to as partial slope estimates. As with simple linear regression, we need to determine the 95% confidence interval for each parameter estimate, so that we could get an idea where the true population parameter lies. Recall from our June 3 post, we did that by determining the equation for the standard error of the estimate, sε, and then the standard error of the regression slope, sb. That worked well for simple regression, but for multiple regression, it is more complicated. Unfortunately, deriving the standard error of the partial regression coefficients requires the use of linear algebra, and would be too complicated to discuss here. Several statistical programs and Excel compute these values for us. So, we will state the values of sb1 and sb2 and go from there.

Sb1=0.05556

Sb2=0.00003

Also, we need our critical-t value for 22 degrees of freedom, which is 2.074.

Hence, our 95% confidence interval for β1 is denoted as:

0.23720 ± 2.074 × 0.05556

=0.12197 to 0.35243

Hence, we are saying that we can be 95% confident that the true parameter β1 lies somewhere between the values of 0.12197 and 0.35243.

Similarly, for β2, the procedure is similar:

-0.000249 ± 2.074 × 0.00003

=-0.00032 to -0.00018

Hence, we can be 95% confident that the true parameter β2 lies somewhere between the values of -0.00032 and -0.00018. Also, the confidence interval for the intercept, α, ranges from 1.40 to 1.73.

Note that in all of these cases, the confidence interval does not contain a value of zero within its range. The confidence intervals for α and β1 are positive; that for β2 is negative. If any parameter’s confidence interval ranges crossed zero, then the parameter estimate would not be significant.

Is Our Model Valid?

The next thing we want to do is determine if our model is valid. When validating our model we are trying to prove that our independent variables explain the variation in the dependent variable. So we start with a hypothesis test:

H0: β1 = β2 = 0

HA: at least one β ≠ 0

Our null hypothesis says that our independent variables, net revenue per deposit dollar and number of S&L offices, explain nothing of the variation in an S&L percentage profit margin, and hence, that our model is not valid. Our alternative hypothesis says that at least one of our independent variable explains some of the variation in an S&L’s percentage profit margin, and thus is valid.

So how do we do it? Enter the F-test. Like the T-test, the F-test is a means for hypothesis testing. Let’s first start by calculating our F-statistic for our model. We do that with the following equation:

Remember that RSS is the regression sum of squares and ESS is the error sum of squares. The May 27th Forecast Friday post showed you how to calculate RSS and ESS. For this model, our RSS=0.4015, and our ESS=0.0625; k is the number of independent variables, and n is the sample. Our equation reduces to:


= 70.66

If our Fcalc is greater than the critical F value for the distribution, then we can reject our null hypothesis and conclude that there is strong evidence that at least one of our independent variables explains some of the variation in an S&L’s percentage profit margin. How do we determine our critical F? There is yet another table in any statistics book or statistics Web site called the “F Distribution” table. In it, you look for two sets of degrees of freedom – one for the numerator and one for the denominator of your Fcalc equation. In the numerator, we have two degrees of freedom; in the denominator, 22. So we look at the F Distribution table notice the columns represent numerator degrees of freedom, and the rows, denominator degrees of freedom. When we find column (2), row (22), we end up with an F-value of 5.72.

Our Fcalc is greater than that, so we can conclude that our model is valid.

Is Our Model Free of Autocorrelation?

Recall from our assumptions that none of our error terms should be correlated with one another. If they are, autocorrelation results, rendering our parameter estimates inefficient. Check for autocorrelation, we need to look at our error terms, when we compare our predicted percentage profit margin, Ŷ, with our actual, Y:

Year

Percentage Profit Margin

Actual (Yt)

Predicted by Model (Ŷt)

Error

1

0.75

0.68

(0.0735)

2

0.71

0.71

0.0033

3

0.66

0.70

0.0391

4

0.61

0.67

0.0622

5

0.7

0.68

(0.0162)

6

0.72

0.71

(0.0124)

7

0.77

0.74

(0.0302)

8

0.74

0.76

0.0186

9

0.9

0.79

(0.1057)

10

0.82

0.79

(0.0264)

11

0.75

0.80

0.0484

12

0.77

0.83

0.0573

13

0.78

0.80

0.0222

14

0.84

0.80

(0.0408)

15

0.79

0.75

(0.0356)

16

0.7

0.73

0.0340

17

0.68

0.70

0.0249

18

0.72

0.69

(0.0270)

19

0.55

0.64

0.0851

20

0.63

0.61

(0.0173)

21

0.56

0.57

0.0101

22

0.41

0.48

0.0696

23

0.51

0.44

(0.0725)

24

0.47

0.40

(0.0746)

25

0.32

0.38

0.0574

The next thing we need to do is subtract the previous period’s error from the current period’s error. After that, we square our result. Note that we will only have 24 observations (we can’t subtract anything from the first observation):

Year

Error

Difference in Errors

Squared Difference in Errors

1

(0.07347)

  

  

2

0.00334

0.07681

0.00590

3

0.03910

0.03576

0.00128

4

0.06218

0.02308

0.00053

5

(0.01624)

(0.07842)

0.00615

6

(0.01242)

0.00382

0.00001

7

(0.03024)

(0.01781)

0.00032

8

0.01860

0.04883

0.00238

9

(0.10569)

(0.12429)

0.01545

10

(0.02644)

0.07925

0.00628

11

0.04843

0.07487

0.00561

12

0.05728

0.00884

0.00008

13

0.02217

(0.03511)

0.00123

14

(0.04075)

(0.06292)

0.00396

15

(0.03557)

0.00519

0.00003

16

0.03397

0.06954

0.00484

17

0.02489

(0.00909)

0.00008

18

(0.02697)

(0.05185)

0.00269

19

0.08509

0.11206

0.01256

20

(0.01728)

(0.10237)

0.01048

21

0.01012

0.02740

0.00075

22

0.06964

0.05952

0.00354

23

(0.07252)

(0.14216)

0.02021

24

(0.07460)

(0.00208)

0.00000

25

0.05738

0.13198

0.01742

 

If we sum up the last column, we will get .1218, if we then divide that by our ESS of 0.0625, we get a value of 1.95. What does this mean?

We have just computed what is known as the Durbin-Watson Statistic, which is used to detect the presence of autocorrelation. The Durbin-Watson statistic, d, can be anywhere from zero to 4. Generally, when d is close to zero, it suggests the presence of positive autocorrelation; a value close to 2 indicates no autocorrelation; while a value close to 4 indicates negative autocorrelation. In any case, you want your Durbin-Watson statistic to be as close to two as possible, and ours is.

Hence, our model seems to be free of autocorrelation.

Now, Let’s Go Forecast!

Now that we have validated our model, and saw that it was free of autocorrelation, we can be comfortable forecasting. Let’s say that for years 26 and 27, we have the following forecasts for net revenues per deposit dollar, X1t and number of S&L offices, X2t. They are as follows:

X1,26 = 4.70 and X2,26 = 9,350

X1,27 = 4.80 and X2,27 = 9,400

Plugging each of these into our equations, we generate the following forecasts:

Ŷ26 = 1.56450 + 0.23720 * 4.70 – 0.000249 * 9,350

=0.3504

Ŷ27 = 1.56450 + 0.23720 * 4.80 – 0.000249 * 9,400

=0.3617

Next Week’s Forecast Friday Topic: The Effect of Omitting an Important Variable

Now that we’ve walked you through this process, you know how to forecast and run multiple regression. Next week, we will discuss what happens when a key independent variable is omitted from a regression model and all the problems it causes when we violate the regression assumption that “all relevant and no irrelevant independent variables are included in the model.” Next week’s post will show a complete demonstration of such an impact. Stay tuned!