*(Twenty-fourth in a series)*

In the last two posts, we discussed the use of dummy variables for factoring the impact of categorical or seasonal phenomena into regression models. Those dummy variables affected the y-intercept of the regression equation. However, many datasets – especially time series – are subject to structural changes that affect the slope – the coefficient – in the regression equation. For example, if you were doing long range forecasting based on several years of data for the airline industry, airline business practices were very different before September 11, 2001 – and you must adjust for it.

Structural changes can also occur in cross-sectional data. If you an operations manager at a factory and were trying to develop a model for worker productivity based on years of experience and education, you might discover that education requirements for factory jobs changed some time ago. Of course, not all current factory workers were affected by the change; some older workers were grandfathered; or union contracts may have shielded these workers from the changes. If for example the newer factory workers were required to obtain a certain amount of college-level training for their work, and you don’t account for the changed requirement, your parameter estimates will be biased.

How do we account for these structural shifts? Slope dummy variables – or slope dummies, for short.

Since the specification of a slope dummy is only slightly more complex than an intercept dummy, I will not be using a full-scale regression example here as I have in past posts. Rather, I will show what a regression model with a slope dummy looks like.

Let’s assume you run a business and your sales are greatly affected by city ordinances – the more ordinances there are, the lower your sales. Your city has two political parties – the Regulation party and the Deregulation party. For the most part, the Regulation party tends to impose more ordinances when they occupy city hall and the Deregulation party tends to impose fewer, or rescind, ordinances when they’re in office.

Of course, sometimes a Regulation administration may not impose new ordinances; and a Deregulation administration may impose them, depending on the policy and economic issues the city is facing at the time. But, for the most part, ordinances tend to increase under Regulation administrations. So how do we account for this?

Let’s start with a simple regression equation:

Ŷ = α -β_{1}O_{t} + ε_{t}

In this equation, Ŷ represents forecasted sales; α is the y-intercept; β_{1} is the parameter estimate for variable O, which is the number of pages of ordinances on the city’s books in that year; ε is the error term; and t is the time period in the regression. Notice that the parameter estimate, β_{1}, is negative, which we would expect. As the number of pages of ordinances increases, we would expect to see sales go down.

But now you want to account for whether the party in office is a Regulation administration. So you create a dummy variable called D_{t}. D_{t}=1 in years when city hall is run by a Regulation mayor and D_{t}=0 when the city is run by a Deregulation mayor. So your new equation looks like this:

Ŷ = α -β_{1}O_{t} – β_{2} O_{t} D_{t }+ ε_{t}

Notice the difference between the slope dummy in this last equation? It’s multiplied against the pages of regulation and then run as an independent variable in the model. Hence, we forecast our sales as follows:

When a Deregulation administration is in city hall (D_{t} =0), Ŷ=

α -β_{1}O_{t};

When a Regulation administration is in city hall (D_{t} =1), Ŷ=

α –(β_{1} +β_{2})O_{t}

Hence, you see the slope (parameter estimate) is different if the Regulation party is in office.

**Next Forecast Friday Topic: Selecting Variables for a Regression Model**

Sometimes you have a lot of variables to choose from when building a regression model. How do you know which ones to include in your model? We will discuss some approaches to determine which variables to enter into your model next week.

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**Check out Analysights’ profile at the Janlong Communications Blog!**

Marketing communications specialist Janice Long of Janlong Communications profiles many small and up-and-coming businesses on her blog, asking owners their ideal client profile and how they got started. This week, Janlong Communications profiled Analysights! See the brief post about how we got started and the market niche we serve.