## Posts Tagged ‘regression’

### Forecast Friday Topic: Correcting Autocorrelation

August 5, 2010

(Sixteenth in a series)

Last week, we discussed how to detect autocorrelation – the violation of the regression assumption that the error terms are not correlated with one another – in your forecasting model. Models exhibiting autocorrelation have parameter estimates that are inefficient, and R2s and t-ratios that seem overly inflated. As a result, your model generates forecasts that are too good to be true and has a tendency to miss turning points in your time series. In last week’s Forecast Friday post, we showed you how to diagnose autocorrelation: examining the model’s parameter estimates, visually inspecting the data, and computing the Durbin-Watson statistic. Today, we’re going to discuss how to correct it.

Revisiting our Data Set

Recall our data set: average hourly wages of textile and apparel workers for the 18 months from January 1986 through June 1987, as reported in the Survey of Current Business (September issues from 1986 and 1987), and reprinted in Data Analysis Using Microsoft ® Excel, by Michael R. Middleton, page 219:

 Month t Wage Jan-86 1 5.82 Feb-86 2 5.79 Mar-86 3 5.8 Apr-86 4 5.81 May-86 5 5.78 Jun-86 6 5.79 Jul-86 7 5.79 Aug-86 8 5.83 Sep-86 9 5.91 Oct-86 10 5.87 Nov-86 11 5.87 Dec-86 12 5.9 Jan-87 13 5.94 Feb-87 14 5.93 Mar-87 15 5.93 Apr-87 16 5.94 May-87 17 5.89 Jun-87 18 5.91

We generated the following regression model:

Ŷ = 5.7709 + 0.0095t

Our model had an R2 of .728, and t-ratios of about 368 for the intercept term and 6.55 for the parameter estimate, t. The Durbin-Watson statistic was 1.05, indicating positive autocorrelation. How do we correct for autocorrelation?

Lagging the Dependent Variable

One of the most common remedies for autocorrelation is to lag the dependent variable one or more periods and then make the lagged dependent variable the independent variable. So, in our data set above, you would take the first value of the dependent variable, \$5.82, and make it the independent variable for period 2, with \$5.79 being the dependent variable; in like manner, \$5.79 will also become the independent variable for the next period, whose dependent variable has a value of \$5.80, and so on. Since the error terms from one period to another exhibit correlation, by using the previous value of the dependent variable to predict the next one, you reduce that correlation of errors.

You can lag for as many periods as you need to; however, note that you lose the first observation when you lag one period (unless you know the previous period before the start of the data set, you have nothing to predict the first observation). You’ll lose two observations if you lag two periods, and so on. If you have a very small data set, the loss of degrees of freedom can lead to Type II error – failing to identify a parameter estimate as significant, when in fact it is. So, you must be careful here.

In this case, by lagging our data by one period, we have the following data set:

 Month Wage Lag1 Wage Feb-86 \$5.79 \$5.82 Mar-86 \$5.80 \$5.79 Apr-86 \$5.81 \$5.80 May-86 \$5.78 \$5.81 Jun-86 \$5.79 \$5.78 Jul-86 \$5.79 \$5.79 Aug-86 \$5.83 \$5.79 Sep-86 \$5.91 \$5.83 Oct-86 \$5.87 \$5.91 Nov-86 \$5.87 \$5.87 Dec-86 \$5.90 \$5.87 Jan-87 \$5.94 \$5.90 Feb-87 \$5.93 \$5.94 Mar-87 \$5.93 \$5.93 Apr-87 \$5.94 \$5.93 May-87 \$5.89 \$5.94 Jun-87 \$5.91 \$5.89

So, we have created a new independent variable, Lag1_Wage. Notice that we are not going to regress time period t as an independent variable. This doesn’t mean that we should or shouldn’t; in this case, we’re only trying to demonstrate the effect of the lagging.

Rerunning the Regression

Now we do our regression analysis. We come up with the following equation:

Ŷ = 0.8253 + 0.8600*Lag1_Wage

Apparently, from this model, each \$1 change in hourly wage from the previous month is associated with an average \$0.86 change in hourly wages for the current month. The R2 for this model was virtually unchanged, 0.730. However, the Durbin-Watson statistic is now 2.01 – just about the total eradication of autocorrelation. Unfortunately, the intercept has a t-ratio of 1.04, indicating it is not significant. The parameter estimate for Lag1_Wage is about 6.37, not much different than the parameter estimate for t in our previous model. However, we did get rid of the autocorrelation.

The statistically insignificant intercept term resulting from this lagging is a result of the Type II error involved with the loss of a degree of freedom in a small sample size. Perhaps if we had several more months of data, we might have had a significant intercept estimate.

Other Approaches to Correcting Autocorrelation

There are other approaches to correcting autocorrelation. One other important way might be to identify important independent variables that have been omitted from the model. Perhaps if we had data on the average years work experience of the textile and apparel labor force from month to month, that might have increased our R2, and reduced correlations in the error term. Another thing we could do is difference the data. Differencing works like lagging, only we subtract the value of the dependent and independent variables of the first observation from their respective values in the second observation; then we subtract those of the second observation’s original values from those of the third, and so on. Then we run a regression on the differences in observations. The problem here is that again, your data set is reduced by one observation and your transformed model will not have an intercept term, which can cause issues in some studies.

Other approaches to correcting autocorrelation include quasi-differencing, the Cochran-Orcutt Procedure, the Hildreth-Lu Procedure, and the Durbin Two-Step Method. These methods are iterative, require a lot of tedious effort and are beyond the scope of our post. But many college-level forecasting textbooks have sections on these procedures if you’re interested in further reading on them.

Next Forecast Friday Topic: Detecting Heteroscedasticity

Next week, we’ll discuss the last of the regression violations, heteroscedasticity, which is the violation of the assumption that error terms have a constant variance. We will discuss why heteroscedasticity exists and how to diagnose it. The week after that, we’ll discuss remedying heteroscedasticity. Once we have completed our discussions on the regression violations, we will spend a couple of weeks discussing regression modeling techniques like transforming independent variables, using categorical variables, adjusting for seasonality, and other regression techniques. These topics will be far less theoretical and more practical in terms of forecasting.

### 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!

### Forecast Friday Topic: Prelude to Multiple Regression Analysis – Regression Assumptions

June 10, 2010

(Eighth in a series)

In last week’s Forecast Friday post, we continued our discussion of simple linear regression analysis, discussing how to check both the slope and intercept coefficients for significance. We then discussed how to create a prediction interval for our forecasts. I had intended this week’s Forecast Friday post to delve straight into multiple regression analysis, but have decided instead to spend some time talking about the assumptions that go into building a regression model.  These assumptions apply to both simple and multiple regression analysis, but their importance is especially noticeable with multiple regression, and I feel it is best to make you aware of them, so that when we discuss multiple regression both as a time series and as a causal/econometric forecasting tool, you’ll know how to detect and correct regression models that violate these assumptions. We will formally begin our discussion of multiple regression methods next week.

Five Key Assumptions for Ordinary Least Squares (OLS) Regression

When we develop our parameter estimates for our regression model, we want to make sure that all of our estimators have the smallest variance. Recall that when you were computing the value of your estimate, b, for the parameter β, in the equation below:

You were subtracting your independent variable’s average from each of its actual values, and doing likewise for the dependent variable. You then multiplied those two quantities together (for each observation) and summed them up to get the numerator of that calculation. To get the denominator, you again subtracted the independent variable’s mean from each of its actual values and then squared them. Then you summed those up. The calculation of the denominator is the focal point here: the value you get for your estimate of β is the estimate that minimizes the squared error for your model. Hence, the term, least squares. If you were to take the denominator of the equation above and divide it by your sample size (less one: n-1), you would get the variance of your independent variable, X. This variance is something you also want to minimize, so that your estimate of β is efficient. When your parameter estimates are efficient, you can make stronger statistical statements about them.

We also want to be sure that our estimators are free of bias. That is, we want to be sure that our sample estimate, b, is on average, equal to our true population parameter, β. That is, if we calculated several estimates of β, the average of our b’s should equal β.

Essentially, there are five assumptions that must be made to ensure our estimators are unbiased and efficient:

Assumption #1: The regression equation correctly specifies the true model.

In order to correctly specify the true model, the relationship between the dependent and independent variable must be linear. Also, we must neither exclude relevant independent variables from nor include irrelevant independent variables in our regression equation. If any of these conditions are not met – that is, Assumption #1 is violated – then our parameter estimates will exhibit bias, particularly specification bias.

In addition, our independent and dependent variables must be measured accurately. For example, if we are trying to estimate salary based on years of schooling, we want to make sure our model is measuring years of schooling as actual years of schooling, and not desired years of schooling.

Assumption #2: The independent variables are fixed numbers and not correlated with error terms.

I warned you at the start of our discussion of linear regression that the error terms were going to be important. Let’s start with the notion of fixed numbers. When you are running a regression analysis, the values of each independent variable should not change every time you test of the equation. That is, the values of your independent variables are known and controlled by you. In addition, the independent variables should not be correlated with the error term. If an independent variable is correlated with the error term, then it is very possible a relevant independent variable was excluded from the equation. If Assumption #2 is violated, then your parameter estimates will be biased.

Assumption #3: The error terms ε, have a mean, or expected value, of zero.

As you noticed in the past blog post, when we developed our regression equation for Sue Stone’s monthly sales, we went back in and plugged each observation’s independent variable into our model and generated estimates of sales for that month. We then subtracted the estimated sales from the actual. Some of our estimates were higher than average, some were lower. Summing up all these errors, they should equal zero. If they don’t, they will result in a biased estimate of the intercept, a (which we use to estimate α). This assumption is not of serious concern, however, since the intercept is often of secondary importance to the slope estimate. We also assume that the error terms are normally distributed.

Assumption #4: The error terms have a constant variance.

The variance of the error term for all values of Xi should be constant, that is, the error terms should be homoscedastic. Visually, if you were to plot the line generated by your regression equation, and then plot the error terms for each observation as points above or below the regression line, the points should cluster around the line in a band of equal width above and below the regression line. If, instead, the points began to move further and further away from the regression line as the value of X increased, then the error terms are heteroscedastic, and the constant variance assumption is violated. Heteroscedasticity does not bias parameter estimates, but makes them inefficient, or untrustworthy.

Why does heteroscedasticity occur? Sometimes, a data set has some observations whose values for the independent variable are vastly different from those of the other observations. These cases are known as outliers. For example, if you have five observations, and their X values were as follows:

{ 5, 6, 6, 7, 20}

The fifth observation would be the outlier, since its X value of 20 is so different from that of the four previous observations. Regression equations place excessive weight on extreme values. Let’s assume that you were trying to construct a model to predict new car purchases based on income. You choose “household income” as your dependent variable and “new car spending” as the dependent variable. You survey 10 people who bought a new car, and you record both their income and the amount they paid for the car. You sort each respondent in order by income and look at their spending, as depicted in the table below:

 Respondent Annual Income New Car Purchase Price 1 \$30,000 \$25,900 2 \$32,500 \$27,500 3 \$35,000 \$26,000 4 \$37,500 \$29,000 5 \$40,000 \$32,000 6 \$42,500 \$30,500 7 \$45,000 \$34,000 8 \$47,500 \$26,500 9 \$50,000 \$38,000 10 \$52,500 \$40,000

Do you notice the pattern that as income increases, the new car purchase price tends to move upward? For the most part, it does. But, does it go up consistently? No. Notice how respondent #3 spent less for a car than the two respondents with lower incomes; respondent #8 spent much less for a car than lower-income respondents 4-7. Respondent #8 is an outlier. This happens because lower-income households are limited in their options for new cars, while higher-income households have more options. A low-income respondent may be limited to buying a Ford Focus or a Honda Civic; but a higher-income respondent may be able to buy a Lexus or BMW, yet still choose to buy the Civic or the Focus. Heteroscedasticity is very likely to occur with this data set. In case you haven’t guessed, heteroscedasticity is more likely to occur with cross-sectional data, rather than with time series data.

Assumption #5: The error terms are not correlated with each other.

Knowing the error term for any of our observations should not allow us to predict the error term of any other observation; the errors must be truly random. If they aren’t, autocorrelation results and the parameter estimates are inefficient, though unbiased. Autocorrelation is much more common with time series data than with cross-sectional data, and occurs because past occurrences can influence future ones. A good example of this is when I was building a regression model to help a college forecast enrollment. I started by building a simple time series regression model, then examined the errors and detected autocorrelation. How did it happen? Because most students who are enrolled in the Fall term are also likely to be enrolled again in the consecutive Spring term. Hence, I needed to correct for that autocorrelation. Similarly, while a company’s advertising expenditures in April may impact its sales in April, they are also likely to have some impact on its sales in May. This too can cause autocorrelation.

When these assumptions are kept, your regression equation is likely to contain parameter estimates that are the “best, linear, unbiased estimators” or BLUE. Keep these in mind as we go through our upcoming discussions on multiple regression.

Next Forecast Friday Topic: Regression with Two or More Independent Variables

Next week, we will plunge into our discussion of multiple regression. I will give you an example of how multiple variables are used to forecast a single dependent variable, and how to check for validity. As we go through the next couple of discussions, I will show you how to analyze the error terms to find violations of the regression assumptions. I will also show you how to determine the validity of the model, and to identify whether all independent variables within your model are relevant.