## Posts Tagged ‘r-squared’

### Forecast Friday Topic: The Logistic Regression Model

November 4, 2010

(Twenty-eighth in a series)

Sometimes analysts need to forecast the likelihood of discrete outcomes: the probability that this outcome will occur or that outcome will occur. Last week, we discussed the linear probability model (LPM) as one solution. Essentially, the LPM looked that the two discrete outcomes: a 1 if the outcome occurred and a 0 if it did not. We treated that binary variable as the dependent variable and then ran it as if it were ordinary least squares. In our example, we got a pretty good result. However, LPM came up short in many different ways: values that fell outside the 0 to 1 range, dubious R2 values, heteroscedasticity, and non-normal error terms.

One of the more popular improvements on the LPM is the logistic regression model, sometimes referred to as the “logit” model. Logit models are very useful in consumer choice modeling. While the theory is quite complex, today we will introduce you to basic concepts of the logit model, using a simple regression model.

Probabilities, Odds Ratios, and Logits, Oh My!

The first thing to understand about logistic regression is the mathematics of the model. There are three calculations that you need to know: the probability, the odds ratio, and the logits. While these are different values, they are three ways of expressing the very same thing: the likelihood of an outcome occurring.

Let’s start with the easiest of the three: probability. Probability is the likelihood that a particular outcome will happen. It is a number between 0 and 1. A probability of .75 means that an outcome has a 75% chance of occurring. In a logit model, the probability of an observation having a “success” outcome (Y=1) is denoted as Pi. Since Pi is a number between 0 and 1, that means the probability of a “failure” outcome (Y=0) is 1- Pi. If Pi (Y=1)=0.40 then Pi (Y=0)=1.00-0.40 = 0.60.

The odds ratio, then, is the ratio of the probability of a success to the probability of failure:

Hence, in the above example, the odds ratio is (0.40/0.60) = 0.667.

Logits, denoted as Li, are the natural log of the odds ratio:

Hence the logit in our example here is ln(0.667) = -0.405.

A logistic regression model’s equation generates Ŷ values in the form of logits for each observation. The logits are equal to the terms of a regression equation:

Once an observation’s logit is generated, you must take its antilog to derive its odds ratio, and then you must use simple algebra to compute the probability.

Estimate the Logit Model Using Weighted Least Squares Regression

Because data with a logistic distribution are not linear, linear regression is often not appropriate for modeling; however, if data can be grouped, ordinary least squares regression can be used to estimate the logits. This example will use a simple one-variable model to approximate logits. Multiple logistic regression is beyond the scope of this post, and should only be used with a statistical package.

We use weighted least squares (WLS) techniques for approximating the logits. You will recall that we discussed WLS in our Forecast Friday post on heteroscedasticity. In a logistic distribution, error terms are heteroscedastic, making WLS an appropriate tool to employ. The steps in this process are:

1. Group the independent variable, each group being its own Xi.
2. Note the sample size, Ni of each group, and count the number of successes, ni, in each.
3. Compute the relative probabilities for each Xi:

1. Use WLS to transform the model with weights, wi:

2. Perform OLS on the weighted, or transformed model:

L*-hat is computed by multiplying L-hat by the weight, w. Likewise X* is computed by multiplying the original X value by weight; similar for error term.

3. Take the antilog of the logits to estimate probabilities for each group.

Predicting Churn in Wireless Telephony

Marissa Martinelli is director of customer retention for Cheapo Wireless, a low-end cell phone provider. Cheapo’s target market are subprime households, whose incomes are generally below \$50,000 per year and don’t have bank accounts. As incomes of their customers rise, churn rates for low-end cell phones increases greatly. Cheapo has developed a new cell phone plan that caters to higher income customers, so that it can migrate its existing customers to the new plan as their incomes rise. In order to promote the new plan, Marissa must first identify the customers most at risk of churning.

Marissa takes a random sample of 1,365 current and former Cheapo cell phone customers and looks at their churn rates. She has their incomes based on their applications and credit checks when they first applied for wireless service. She decides to break them down into 19 groups, with incomes from \$0 to \$50,000, in \$2,500 increments. For simplicity, Marissa divides the income amounts by \$10,000, and decides to group them. The lowest income group, 0.50, is all customers whose incomes are \$5,000 or less; the next group, 0.75, are those with incomes between \$5,000-\$7,500, and so on. Marissa notes the number of churned customers (ni) for each income level and the number of customers for each income level (Ni):

 # Churned Group Size Income level (\$10Ks) ni Ni Xi 1 20 0.50 2 30 0.75 3 30 1.00 5 40 1.25 6 40 1.50 8 50 1.75 9 50 2.00 12 60 2.25 17 80 2.50 22 80 2.75 35 100 3.00 40 100 3.25 75 150 3.50 70 125 3.75 62 100 4.00 62 90 4.25 64 90 4.50 51 70 4.75 50 60 5.00

As the table shows, of the 60 customers whose income is between \$47,500 and \$50,000, 50 of them have churned. Knowing this information, Marissa can now compute the conditional probabilities of churn (Y=1) for each income group:

 # Churned Group Size Income level (\$10Ks) Probability of Churn Probability of Retention ni Ni Xi Pi 1-Pi 1 20 0.50 0.050 0.950 2 30 0.75 0.067 0.933 3 30 1.00 0.100 0.900 5 40 1.25 0.125 0.875 6 40 1.50 0.150 0.850 8 50 1.75 0.160 0.840 9 50 2.00 0.180 0.820 12 60 2.25 0.200 0.800 17 80 2.50 0.213 0.788 22 80 2.75 0.275 0.725 35 100 3.00 0.350 0.650 40 100 3.25 0.400 0.600 75 150 3.50 0.500 0.500 70 125 3.75 0.560 0.440 62 100 4.00 0.620 0.380 62 90 4.25 0.689 0.311 64 90 4.50 0.711 0.289 51 70 4.75 0.729 0.271 50 60 5.00 0.833 0.167

Marissa then goes on to derive the weights for each income level:

 Logits Weights Pi *(1-Pi) Pi /(1-Pi) Li NiPi(1-Pi) Wi 0.048 0.053 -2.944 0.950 0.975 0.062 0.071 -2.639 1.867 1.366 0.090 0.111 -2.197 2.700 1.643 0.109 0.143 -1.946 4.375 2.092 0.128 0.176 -1.735 5.100 2.258 0.134 0.190 -1.658 6.720 2.592 0.148 0.220 -1.516 7.380 2.717 0.160 0.250 -1.386 9.600 3.098 0.167 0.270 -1.310 13.388 3.659 0.199 0.379 -0.969 15.950 3.994 0.228 0.538 -0.619 22.750 4.770 0.240 0.667 -0.405 24.000 4.899 0.250 1.000 0.000 37.500 6.124 0.246 1.273 0.241 30.800 5.550 0.236 1.632 0.490 23.560 4.854 0.214 2.214 0.795 19.289 4.392 0.205 2.462 0.901 18.489 4.300 0.198 2.684 0.987 13.843 3.721 0.139 5.000 1.609 8.333 2.887

Now, Marissa must transform the logits and the independent variable (Income level) by multiplying them by their respective weights:

 Income level (\$10Ks) Logits Weights Weighted Income Weighted Logits Xi Li Wi Xi* Li* 0.50 -2.944 0.975 0.487 -2.870 0.75 -2.639 1.366 1.025 -3.606 1.00 -2.197 1.643 1.643 -3.610 1.25 -1.946 2.092 2.615 -4.070 1.50 -1.735 2.258 3.387 -3.917 1.75 -1.658 2.592 4.537 -4.299 2.00 -1.516 2.717 5.433 -4.119 2.25 -1.386 3.098 6.971 -4.295 2.50 -1.310 3.659 9.147 -4.793 2.75 -0.969 3.994 10.983 -3.872 3.00 -0.619 4.770 14.309 -2.953 3.25 -0.405 4.899 15.922 -1.986 3.50 0.000 6.124 21.433 0.000 3.75 0.241 5.550 20.812 1.338 4.00 0.490 4.854 19.415 2.376 4.25 0.795 4.392 18.666 3.491 4.50 0.901 4.300 19.349 3.873 4.75 0.987 3.721 17.673 3.674 5.00 1.609 2.887 14.434 4.646

Now, Marissa can run OLS on the transformed model, using Weights (wi) and Weighted Income (X*) as independent variables and the Weighted Logits (L*) as the dependent variable.

Marissa derives the following regression equation:

Interpreting the Model

As expected, weighted income has a positive relationship on likelihood of churn. However, her sample is just 19 observations, so Marissa must be very careful about drawing too strong an inference from these results. While R2 is a strong 0.981, it too must not be relied upon. In fact, it is pretty meaningless in a logit model. Also, notice that there is no intercept term in this model. You will recall that when using WLS to correct for heteroscedasticity, the intercept was lost in the transformed model and actually became its slope. It is equivalent to the slope in an unadjusted regression model, since heteroscedasticity doesn’t bias parameter estimates.

Calculating Probabilities

Now Marissa needs to use this model to assess current customers’ likelihood of churning. Let’s say she sees a customer who makes \$9,500 a year. That customer would be in the income group, 1.0. What is that customer’s probability of churning? Marissa takes the weight, 1.643 for her wi and the weighted X* (also 1.643), and plugs them into her equation:

= -4.121

To get to the probability, Marissa must take the antilog of these logits, which will give her the odds ratio: 0.016

Now Marissa calculates this customer’s probability of churning:

So, a customer earning \$9,500 per year has less than a two percent chance of churning. Had the customer been earning \$46,000, he/she would have had a whopping 98.7% chance of churning!

There are equivalents of R2 that are used for logistic regression, but that discussion is beyond the scope of this post. Today’s post was to give you a primer on the theory of logistic regression.

Next Forecast Friday Topic: The Identification Problem

We have just concluded our discussions on qualitative choice models. Next week, we begin our three-part miniseries on Simultaneous Equations and Two-Stage Least Squares Regression. The first post will discuss the Identification problem.

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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: Building Regression Models With Excel

July 8, 2010
(Twelfth in a series)
We’ve spent the last six weeks discussing regression analysis as a forecasting method. As you have seen, simple regression is a bit tedious to work out by hand, but for multiple regression analysis, you almost always need the aid of a computerized software package. Today I will demonstrate for you how to use the Regression Analysis feature of Microsoft Excel’s1 Analysis ToolPak Add-In. Excel 2007 comes with the Analysis ToolPak Add-In, which you can choose to activate. One way to know if Analysis ToolPak is activated on your version of Excel is to click on the Data tab on your workspace and see if there is a “Data Analysis” icon. The following thumbnail will illustrate:

Checking for the Data Analysis Add-In

Notice towards the upper right corner of the image, “Data Analysis” is highlighted in orange. The presence of the Data Analysis icon means that we have activated the Analysis ToolPak Add-In. If it wasn’t there, you would need to activate the Add-In, which you could do very easily by clicking the “Office” button in the top left hand corner, then clicking the “Excel Options” button, which will take you through the process of activating your add-in.

Setting up a Regression – Our Data Set

Generally in direct mail marketing, three components that often determine how much one spends – or whether he/she buys at all – are recency, frequency, and monetary value – known in short as RFM. Generally, the longer it has been since one’s last purchase (recency), the less he/she is likely to spend. Hence, we would expect a minus sign by the coefficient for recency. Also, RFM theorizes that the more frequently one buys, the greater his/her purchase. So, we would expect a plus sign by the coefficient for frequency. Finally, the higher the customer’s average purchases (monetary value), the greater his/her spending, so we would also expect a plus sign here. RFM is also used heavily by nonprofits in their capital and contributor campaigns, since they are often heavily reliant upon direct mail.

In our example here, a local nonprofit decided to test whether each RFM component had a relationship to a donor’s contribution, so it randomly selected 20 donors who contributed to its last appeal. Naturally, our dependent variable, Y, was the Contribution amount. The nonprofit also looked at three independent variables: months since last contribution (X1), times donated in last 12 months (X2), and average contribution over the last 12 months (X3). These independent variables represent recency, frequency, and monetary value, respectively. The table below shows our dataset:

 Giving Patterns of 20 Donors Donor Contribution Months Since Last Donation Times Donated in last 12 months Average Contribution in last 12 months 1 95 10 1 85 2 110 8 2 95 3 100 10 2 90 4 115 8 3 75 5 100 9 1 95 6 120 6 2 100 7 105 9 1 90 8 125 10 1 125 9 105 9 2 100 10 130 4 3 150 11 135 7 4 125 12 150 2 8 150 13 140 4 3 125 14 155 2 9 140 15 140 2 8 130 16 160 2 10 150 17 145 3 6 135 18 165 1 12 150 19 150 3 4 160 20 170 1 12 140

The thumbnail below shows what the data set looks like in Excel:

Regression Data Set in Excel

Running the Regression

To run the regression, we need to select the regression tool from the Analysis ToolPak. We do this by clicking on the Data Analysis Tab. The  next thumbnail shows us what we need to do:

Selecting the Regression option from the Data Analysis ToolPak

After selecting the regression tool, we need to select our independent variables and our dependent variables. It is best to make sure all columns containing your independent variables are adjacent to each other, as they are in columns D, E, and F. Notice that column C from rows 2 to 22 contains our Y-range values (including the column label). In columns D, E, and F, rows 2 through 22 contain their respective X-range values. Notice in the thumbnail how we indicate those column/row positions for Y-range and X-range values.

Regression Options

We also need to decide where to place the regression output and what data we want the output to contain. In the thumbnail below, we choose to have the output placed in a new worksheet, called “Regression Output”, and we also check the box indicating that we want the residuals printed. Also notice that we checked the box “Labels”, so that row 2 won’t be inadvertently added into the model.

Regression Options - Continued

Looking at the Output

Now we run the regression and get the following output:

Regression Output (residuals not shown)

As you can see, cell B5 contains our R2, equal to .933, indicating that 93.3% of the variation in a donor’s contribution amount is explained by changes in recency, frequency, and monetary value. Also, notice the F-statistic in cell E12. It’s a large, strong 73.90, and cell F12 to the right is 0.00, suggesting the model is significant. (Note, the Significance F in cell F12 and the P-Values in cells E17-E20 for each parameter estimate are quick cues to significance. If you’re using a 95% confidence interval – which we are here – then you want those values to be no higher than 0.05).

Now let’s look at each parameter estimate. Cells B17-B20 contain our regression coefficients. We have the following equation:

Contribution Estimate = 87.27 – 1.80 *Months_Since_Last_Donation + 2.45 *Times_Donated_Last_12_Months + 0.35 *Average_Contribution_Last_12_Months

Simplifying, we have:

Contribution Estimate = 87.27 – 1.80*RECENCY + 2.45*FREQUENCY + 0.35*MONETARY_VALUE

Ŷ = 87.27 – 1.80X1 + 2.45X2 + 0.35X3

Note that even though we opted to display the residuals for each observation, I chose not to show them here.  It would have run below the fold, and would have been difficult to see.  Besides, for our analysis, we’re not going to worry about residuals right now.

Interpreting the Output

As we can see, each month since a donor’s last contribution reduces his contribution by an average of \$1.80, when we hold frequency and monetary value constant. Likewise, for each time a donor has given in the last 12 months, the size of his contribution increases by an average of \$2.45, holding the other two variables constant. In addition, each one-dollar increase in a donor’s average contribution increases his contribution by an average of 35 cents. Hence, all of our coefficients have the signs we expect.

T-Statistics and P-Values

Next, we need to look at the t-statistics and P-values. As mentioned above, for a 95% confidence interval, a parameter estimate must have a p-value no greater than 0.05 (or 0.10 for a 90% confidence interval, etc.), in order to be significant. In like manner, for a 95% confidence interval, t-statistics should be values of at least 1.96 (slightly higher for small samples, but 1.96 will work) or less than -1.96 if the coefficient is negative, to be 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

Notice that the coefficient for Months Since Last Donation has a t-statistic of -1.44. It is not significant. Another way to tell whether the parameter estimates are significant is to look at the Lower 95% and Upper 95% values in columns F and G.  If the lower and upper 95% confidence interval values for a parameter estimate are both negative or both positive, they are significant.  However, if the lower 95% value is negative and the upper 95% is positive (as is the case with Months Since Last Donation), then the parameter estimate is not significant, since its confidence interval range crosses zero.  Hence Months Since Last Donation is not significant.  Yet, the model still has a 93.3% coefficient of determination. Does this mean we can drop this variable from our regression? Not so fast!

Regression Violation Present!

Generally, when an independent variable we expect to be an important predictor of our dependent variable comes up as statistically insignificant, it is sometimes a sign of multicollinearity. And that is definitely the case with the nonprofit’s model. That will be our topic in next week’s Forecast Friday post.

Forecasting with the Output

Since we’re going to take on multicollinearity next week, let’s pretend our model is A-OK, and generate some forecasts.

We’ll go to our regression output worksheet, select cells A17 through B20, which contain our regression variables and coefficients, and then click Copy (or do a CTRL-C):

Selecting the Coefficients

Next, let’s paste those coefficients and transpose them in another worksheet.  Here’s how to select the “Transpose” option when pasting:

Pasting Data Using the Transpose Option

Next, this is what the result of our transpose will be:

Transposed Data

Now, the nonprofit organization looks at five prospective donors whom they are planning to solicit. They look at their past giving history as shown in the next thumbnail:

Prospective Donors - Before Applying Model

Knowing this information, we want to multiply those values by their respective coefficients. Take a look at the formula in cell F7 as we do just that, in the next thumbnail:

Forecasting with Regression Output

Note how the cell numbers containing the coefficients have their column letters enveloped in ‘\$’. The dollar signs tell Excel that when we copy the formula down the next four rows, that it still reference those cells. Otherwise, for each row down, Excel would multiply each blank cell below the coefficients by the next donor’s information.  Here’s are the forecasts generated:

Next Forecast Friday Topic: Multicollinearity

Today you learned how to develop regression models using Excel and how to use Excel to interpret the output. You also found out that our model exhibited multicollinearity, a violation of one of the key regression assumptions. Next week and the week after, we will discuss multicollinearity in depth: how to detect it, how to correct it, and when to live with it. We will again be using the nonprofit’s model.  As I’ve said before, models are far from perfect and, as such, should only aid – not replace – the decision-making process.

1 Note: Excel is a registered trademark of Microsoft Corporation. Use of Microsoft Excel in this post is intended only for a demonstration of how to use Excel for regression analysis and does not constitute an endorsement of Microsoft Excel or any other Microsoft product by Analysights, LLC.

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### Multiple Regression: Specification Bias

July 1, 2010

(Eleventh in a series)

In last week’s Forecast Friday post, we discussed several of the important checks you must do to ensure that your model is valid. You always want to be sure that your model does not violate the assumptions we discussed earlier. Today we are going to see what happens when we violate the specification assumption, which says that we do not omit relevant independent variables from our regression model. You will see that when we leave out an important independent variable from a regression model, quite misleading results can emerge. You will also see that violating one assumption can trigger violations of other assumptions.

Revisiting our Multiple Regression Example

Recall our data set of 25 annual observations of U.S. Savings and Loan profit margin data, shown in the table below:

 Year Percentage Profit Margin (Yt) Net Revenues Per Deposit Dollar (X1t) Number of Offices (X2t) 1 0.75 3.92 7,298 2 0.71 3.61 6,855 3 0.66 3.32 6,636 4 0.61 3.07 6,506 5 0.70 3.06 6,450 6 0.72 3.11 6,402 7 0.77 3.21 6,368 8 0.74 3.26 6,340 9 0.90 3.42 6,349 10 0.82 3.42 6,352 11 0.75 3.45 6,361 12 0.77 3.58 6,369 13 0.78 3.66 6,546 14 0.84 3.78 6,672 15 0.79 3.82 6,890 16 0.70 3.97 7,115 17 0.68 4.07 7,327 18 0.72 4.25 7,546 19 0.55 4.41 7,931 20 0.63 4.49 8,097 21 0.56 4.70 8,468 22 0.41 4.58 8,717 23 0.51 4.69 8,991 24 0.47 4.71 9,179 25 0.32 4.78 9,318

Data taken from Spellman, L.J., “Entry and profitability in a rate-free savings and loan market.” Quarterly Review of Economics and Business, 18, no. 2 (1978): 87-95, Reprinted in Newbold, P. and Bos, T., Introductory Business & Economic Forecasting, 2nd Edition, Cincinnati (1994): 136-137

Also, recall that we built a model that hypothesized that S&L percentage profit margin (our dependent variable, Yt) was positively related to net revenues per deposit dollar (one of our independent variables, X1t), and negatively related to the number of S&L offices (our other independent variable, X2t). When we ran our regression, we got the following model:

Yt = 1.56450 + 0.23720X1t – 0.000249X2t

We also checked to see if the model parameters were significant, and obtained the following information:

 Parameter Value T-Statistic Significant? Intercept 1.5645000 19.70 Yes B1t 0.2372000 4.27 Yes B2t (0.0002490) (7.77) Yes

We also had a coefficient of determination – R2 – of 0.865, indicating that the model explains about 86.5% of the variation in S&L percentage profit margin.

Welcome to the World of Specification Bias…

Let’s deliberately leave out the number of S&L offices (X2t) from our model, and do just a simple regression with the net revenues per deposit dollar. This is the model we get:

Yt = 1.32616 – 0.16913X1t

We also get an R2 of 0.495. The t-statistics for our intercept and parameter B1t are as follows:

 Parameter Value T-Statistic Significant? Intercept 1.32616 9.57 Yes B1t (0.16913) (4.75) Yes

Compare these new results with our previous results and what do you notice? The results of our second regression are in sharp contrast to those of our first regression. Our new model has far less explanatory power – R2 dropped from 0.865 to 0.495 – and the sign of the parameter estimate for net revenue per deposit dollar has changed: The coefficient of X1t was significant and positive in the first model, and now it is significant and negative! As a result, we end up with a biased regression model.

… and to the Land of Autocorrelation…

Recall another of the regression assumptions: that error terms should not be correlated with one another. When error terms are correlated with one another, we end up with autocorrelation, which renders our parameter estimates inefficient. Recall that last week, we computed the Durbin-Watson test statistic, d, which is an indicator of autocorrelation. It is bad to have either positive autocorrelation (d close to zero), or negative autocorrelation (d close to 4). Generally, we want d to be approximately 2. In our first model, d was 1.95, so autocorrelation was pretty much nonexistent. In our second model, d=0.85, suggesting the presence of significant positive autocorrelation!

How did this happen? Basically, when an important variable is omitted from regression, its impact on the dependent variable gets incorporated into the error term. If the omitted independent variable is correlated with any of the included independent variables, the error terms will also be correlated.

…Which Leads to Yet Another Violation!

The presence of autocorrelation in our second regression reveals the presence of another violation, not in the incomplete regression, but in the full regression. As the sentence above read: “if the independent variable is correlated with any of the included independent variables…” Remember the other assumption: “no linear relationship between two or more independent variables?” Basically, the detection of autocorrelation in the incomplete regression revealed that the full regression violated this very assumption – and thus exhibits multicollinearity! Generally, a coefficient changing between positive and negative (either direction) when one or more variables is omitted is an indicator of multicollinearity.

So was the full regression wrong too? Not terribly. As you will find in upcoming posts, avoiding multicollinearity is nearly impossible, especially with time series data. That’s because multicollinearity is typically a data problem. The severity of multicollinearity can often be reduced by increasing the number of observations in the data set. This is often not a problem with cross-sectional data, where data sets can have thousands, if not millions of observations. However, with time series data, the number of observations available is limited to how many periods of data have been recorded.

Moreover, the longer your time series, the more you risk structural changes in your data over the course of your time series. For instance, if you were examining annual patterns in bank lending within a particular census tract between 1990 and 2010, you might have a reliable model to work with. But let’s say you widen your time series to go back as far as 1970. You will see dramatic shifts in patterns in your data set. That’s because prior to 1977, when Congress passed the Community Reinvestment Act, many banks engaged in a practice called “redlining,” where they literally drew red lines around some neighborhoods, usually where minorities and low-income households were, and did not lend there. In this case, increasing the size of the data set might reduce multicollinearity, but actually cause other modeling problems.

And as you’ve probably guessed, one way of reducing multicollinearity can be dropping variables from the regression. But look what happened when we dropped the number of S&L offices from our regression: we might have eliminated multicollinearity, but we gained autocorrelation and specification bias!

Bottom Line:

The lesson, for us as forecasters and analysts, therefore is that we must accept that models are far from perfect and we must weigh the impact of various regression model specifications. Is the multicollinearity that is present in our model tolerable? Can we add more observations without causing new problems? Can we drop a variable from a regression without causing either specification bias or material differences in explanatory power, parameter estimates, model validity, or even forecast accuracy? Building the model is easy – but it’s these normative considerations that’s challenging.

Next Forecast Friday Topic: Building Regression Models Using Excel

In next week’s Forecast Friday post, we will take a break from discussing the theory of regression analysis and look at a demonstration of how to use the “Regression Analysis” tool in Microsoft Excel. This demonstration is intended to show you how easy running a regression is, so that you can start applying the concepts and building forecasts for your business. Until then, thanks again for reading Forecast Friday, and I wish you and your family a great 4th of July weekend!

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### Forecast Friday Topic: Multiple Regression Analysis

June 17, 2010

(Ninth in a series)

Quite often, when we try to forecast sales, more than one variable is often involved. Sales depends on how much advertising we do, the price of our products, the price of competitors’ products, the time of the year (if our product is seasonal), and also demographics of the buyers. And there can be many more factors. Hence, we need to measure the impact of all relevant variables that we know drive our sales or other dependent variable. That brings us to the need for multiple regression analysis. Because of its complexity, we will be spending the next several weeks discussing multiple regression analysis in easily digestible parts. Multiple regression is a highly useful technique, but is quite easy to forget if not used often.

Another thing to note, regression analysis is often used for both time series and cross-sectional analysis. Time series is what we have focused on all along. Cross-sectional analysis involves using regression to analyze variables on static data (such as predicting how much money a person will spend on a car based on income, race, age, etc.). We will use examples of both in our discussions of multiple regression.

Determining Parameter Estimates for Multiple Regression

When it comes to deriving the parameter estimates in a multiple regression, the process gets both complicated and tedious, even if you have just two independent variables. We strongly advise you to use the regression features of MS-Excel, or some statistical analysis tool like SAS, SPSS, or MINITAB. In fact, we will not work out the derivation of the parameters with the data sets, but will provide you the results. You are free to run the data we provide on your own to replicate the results we display. I do, however, want to show you the equations for computing the parameter estimates for a three-variable (two independent variables and one dependent variable), and point out something very important.

Let’s assume that sales is your dependent variable, Y, and advertising expenditures and price are your independent variables, X1 and X2, respectively. Also, the coefficients – your parameter estimates will have similar subscripts to correspond to their respective independent variable. Hence, your model will take on the form:

Now, how do you go about computing α, β1 and β2? The process is similar to that of a two-variable model, but a little more involved. Take a look:

The subscript “i” represents the individual oberservation.  In time series, the subscript can also be represented with a “t“.

What do you notice about the formulas for computing β1 and β2? First, you notice that the independent variables, X1 and X2, are included in the calculation for each coefficient. Why is this? Because when two or more independent variables are used to estimate the dependent variable, the independent variables themselves are likely to be related linearly as well. In fact, they need to be in order to perform multiple regression analysis. If either β1 or β2 turned out to be zero, then simple regression would be appropriate. However, if we omit one or more independent variables from the model that are related to those variables in the model, we run into serious problems, namely:

Specification Bias (Regression Assumptions Revisited)

Recall from last week’s Forecast Friday discussion on regression assumptions that 1) our equation must correctly specify the true regression model, namely that all relevant variables and no irrelevant variables are included in the model and 2) the independent variables must not be correlated with the error term. If either of these assumptions is violated, the parameter estimates you get will be biased. Looking at the above equations for β1 and β2, we can see that if we excluded one of the independent variables, say X2, from the model, the value derived for β1 will be incorrect because X1 has some relationship with X2. Moreover, X2‘s values are likely to be accounted for in the error terms, and because of its relationship with X1, X1 will be correlated with the error term, violating the second assumption above. Hence, you will end up with incorrect, biased estimators for your regression coefficient, β1.

Omitted Variables are Bad, but Excessive Variables Aren’t Much Better

Since omitting relevant variables can lead to biased parameter estimates, many analysts have a tendency to include any variable that might have any chance of affecting the dependent variable, Y. This is also bad. Additional variables means that you need to estimate more parameters, and that reduces your model’s degrees of freedom and the efficiency (trustworthiness) of your parameter estimates. Generally, for each variable – both dependent and independent – you are considering, you should have at least five data points. So, for a model with three independent variables, your data set should have 20 observations.

Another Important Regression Assumption

One last thing about multiple regression analysis – another assumption, which I deliberately left out of last week’s discussion, since it applies exclusively to multiple regression:

No combination of independent variables should have an exact linear relationship with one another.

OK, so what does this mean? Let’s assume you’re doing a model to forecast the effect of temperature on the speed at which ice melts. You use two independent variables: Celsius temperature and Fahrenheit temperature. What’s the problem here? There is a perfect linear relationship between these two variables. Every time you use a particular value of Fahrenheit temperature, you will get the same value of Celsius temperature. In this case, you will end up with multicollinearity, an assumption violation that results in inefficient parameter estimates. A relationship between independent variables need not be perfectly linear for multicollinearity to exist. Highly correlated variables can do the same thing. For example, independent variables such as “Husband Age” and “Wife Age,” or “Home Value” and “Home Square Footage” are examples of independent variables that are highly correlated.

You want to be sure that you do not put variables in the model that need not be there, because doing so could lead to multicollinearity.

Now Can We Get Into Multiple Regression????

Wasn’t that an ordeal? Well, now the fun can begin! I’m going to use an example from one of my old graduate school textbooks, because it’s good for several lessons in multiple regression. This data set is 25 annual observations to predict the percentage profit margin (Y) for U.S. savings and loan associations, based on changes in net revenues per deposit dollar (X1) and number of offices (X2). The data are as follows:

 Year Percentage Profit Margin (Yt) Net Revenues Per Deposit Dollar (X1t) Number of Offices (X2t) 1 0.75 3.92 7,298 2 0.71 3.61 6,855 3 0.66 3.32 6,636 4 0.61 3.07 6,506 5 0.70 3.06 6,450 6 0.72 3.11 6,402 7 0.77 3.21 6,368 8 0.74 3.26 6,340 9 0.90 3.42 6,349 10 0.82 3.42 6,352 11 0.75 3.45 6,361 12 0.77 3.58 6,369 13 0.78 3.66 6,546 14 0.84 3.78 6,672 15 0.79 3.82 6,890 16 0.70 3.97 7,115 17 0.68 4.07 7,327 18 0.72 4.25 7,546 19 0.55 4.41 7,931 20 0.63 4.49 8,097 21 0.56 4.70 8,468 22 0.41 4.58 8,717 23 0.51 4.69 8,991 24 0.47 4.71 9,179 25 0.32 4.78 9,318

Data taken from Spellman, L.J., “Entry and profitability in a rate-free savings and loan market.” Quarterly Review of Economics and Business, 18, no. 2 (1978): 87-95, Reprinted in Newbold, P. and Bos, T., Introductory Business & Economic Forecasting, 2nd Edition, Cincinnati (1994): 136-137

What is the relationship between the S&Ls’ profit margin percentage and the number of S&L offices? How about between the margin percentage and the net revenues per deposit dollar? Is the relationship positive (that is, profit margin percentage moves in the same direction as its independent variable(s))? Or negative (the dependent and independent variables move in opposite directions)? Let’s look at each independent variable’s individual relationship with the dependent variable.

Net Revenue Per Deposit Dollar (X1) and Percentage Profit Margin (Y)

Generally, if revenue per deposit dollar goes up, would we not expect the percentage profit margin to also go up? After all, if the S & L is making more revenue on the same dollar, it suggests more efficiency. Hence, we expect a positive relationship. So, in the resulting regression equation, we would expect the coefficient, β1, for net revenue per deposit dollar to have a “+” sign.

Number of S&L Offices (X2) and Percentage Profit Margin (Y)

Generally, if there are more S&L offices, would that not suggest either higher overhead, increased competition, or some combination of the two? Those would cut into profit margins. Hence, we expect a negative relationship. So, in the resulting regression equation, we would expect the coefficient, β2, for number of S&L offices to have a “-” sign.

Are our Expectations Correct?

Do our relationship expectations hold up?  They certainly do. The estimated multiple regression model is:

Yt = 1.56450 + 0.23720X1t – 0.000249X2t

What do the Parameter Estimates Mean?

Essentially, the model says that if net revenues per deposit dollar (X1t) increase by one unit, then percentage profit margin (Yt) will – on average – increase by 0.23720 percentage points, when the number of S&L offices is fixed. If the number of offices (X2t) increases by one, then percentage profit margin (Yt) will decrease by an average of 0.000249 percentage points, when net revenues are fixed.

Do Changes in the Independent Variables Explain Changes in The Dependent Variable?

We compute the coefficient of determination, R2, and get 0.865, indicating that changes in the number of S&L offices and in the net revenue per deposit dollar explain 86.5% of the variation in S&L percentage profit margin.

Are the Parameter Estimates Statistically Significant?

We have 25 observations, and three parameters – two coefficients for the independent variables, and one intercept – hence we have 22 degrees of freedom (25-3). If we choose a 95% confidence interval, we are saying that if we resampled and replicated this analysis 100 times, the average of our parameter estimates will be contain the true parameter approximately 95 times. To do this, we need to look at the t-values for each parameter estimate. For a two-tailed 95% significance test with 22 degrees of freedom, our critical t-value is 2.074. That means that if the t-statistic for a parameter estimate is greater than 2.074, then there is a strong positive relationship between the independent variable and the dependent variable; if the t-statistic for the parameter estimate is less than -2.074, then there is a strong negative relationship. This is what we get:

 Parameter Value T-Statistic Significant? Intercept 1.5645000 19.70 Yes B1t 0.2372000 4.27 Yes B2t (0.0002490) (7.77) Yes

So, yes, all our parameter estimates are significant.

Next Forecast Friday: Building on What You Learned

I think you’ve had enough for this week! But we are still not finished. We’re going to stop here and continue with further analysis of this example next week. Next week, we will discuss computing the 95% confidence interval for the parameter estimates; determining whether the model is valid; and checking for autocorrelation. The following Forecast Friday (July 1) blog post will discuss specification bias in greater detail, demonstrating the impact of omitting a key independent variable from the model.