Posts Tagged ‘Survey of Current Business’

Forecast Friday Topic: Leading Indicators and Surveys of Expectations

December 9, 2010

(Thirty-second in a series)

Most of the forecasting methods we have discussed so far deal with generating forecasts for a steady-state scenario. Yet the nature of the business cycle is such that there are long periods of growth, long periods of declines, and periods of plateau. Many managers and planners would love to know how to spot the moment when things are about to change for better or worse. Spotting these turning points can be difficult given standard forecasting procedures; yet being able to identify when business activity is going to enter a prolonged period of expansion or a protracted decline can greatly enhance managerial and organizational planning. Two of the most common ways managers anticipate turning points in a time series include leading economic indicators and surveys of expectations. This post discusses both.

Leading Economic Indicators

Nobody has a crystal ball. Yet, some time series exhibit patterns that foreshadow economic activity to come. Quite often, when activity turns positive in one time series, months later it triggers an appropriate response in the broader economy. When movements in a time series seem to anticipate coming economic activity, the time series is said to be a leading economic indicator. When a time series moves in tandem with economic activity, the time series is said to be a coincident economic
indicator; and when movements within a particular time series trails economic activity, the time series is said to be a lagging indicator. Economic indicators are nothing new. The ancient Phoenicians, whose empire was built on trading, often used the number of ships arriving in port as an indicator of trading and economic activity.

Economic indicators can be procyclic – that is they increase as economic activity increases and decrease when economic activity decreases; or countercyclic – meaning they decline when the economy is improving or increase when the economy is declining; or they can be acyclic, having little or no correlation at all with the broader economy. Acyclic indicators are rare, and usually are relegated to subsectors of the economy, to which they are either procyclic or countercyclic.

Since 1961, the U.S. Department of Commerce has published the Survey of Current Business, which details monthly changes in leading indicators. The Conference Board publishes a composite index of 10 leading economic indicators, whose activity suggests changes in economic activity six to nine months into the future. Those 10 components include (reprinted from Investopedia.com):

  1. the average weekly hours worked by manufacturing workers;
  2. the average number of initial applications for unemployment insurance;
  3. the amount of manufacturers’ new orders for consumer goods and materials;
  4. the speed of delivery of new merchandise to vendors from suppliers;
  5. the amount of new orders for capital goods unrelated to defense;
  6. the amount of new building permits for residential buildings;
  7. the S&P 500 stock index;
  8. the inflation-adjusted monetary supply (M2);
  9. the spread between long and short interest rates; and
  10. consumer sentiment

 

These indicators are used to measure changes in the broader economy. Each industry or organization may have its own indicators of business activity. For your business, the choice of the time series(‘) to use as leading indicators and the weight they receive depend on several factors, including:

  1. How well it tends to lead activity in your firm and industry;
  2. How easy the time series is to measure accurately;
  3. How well it conforms to the business cycle;
  4. The time series’ overall performance, not just turning points;
  5. Smoothness – no random blips that give misleading economic cues; and
  6. Availability of data.

Over time, the use of specific indicators, and their significance in forecasting do in fact change. You need to keep an eye on how well the indicators you select continue to foreshadow business activity in your industry.

Surveys of Expectations

Sometimes time series are not available for economic indicators. Changes in technology social structure may not be readily picked up in the existing time series. Other times, consumer sentiment isn’t totally represented in the economic indicators. As a result, surveys are used to measure business optimism, or expectations of the future. Economists and business leaders are often surveyed for their opinions. Sometimes, it’s helpful to know if business leaders anticipate spending more money on equipment purchases in the coming year; whether they plan to hire or lay off workers; or whether they intend to expand. While what respondents to these surveys say and what they really do can be quite different, overall, the surveys can provide some direction as to which way the economy is heading.

Next Forecast Friday Topic: Calendar Effects in Forecasting

Easter can fall in March or April; every four years, February has an extra day; in some years, months have four weekends; others years, five. These nuances can generate huge forecast errors. Next week’s Forecast Friday post discusses these calendar effects in forecasting and what you can do to adjust for them.

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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: Detecting Autocorrelation

July 29, 2010

(Fifteenth in a series)

We have spent the last few Forecast Friday posts discussing violations of different assumptions in regression analysis. So far, we have discussed the effects of specification bias and multicollinearity on parameter estimates, and their corresponding effect on your forecasts. Today, we will discuss another violation, autocorrelation, which occurs when sequential residual (error) terms are correlated with one another.

When working with time series data, autocorrelation is the most common problem forecasters face. When the assumption of uncorrelated residuals is violated, we end up with models that have inefficient parameter estimates and upwardly-biased t-ratios and R2 values. These inflated values make our forecasting model appear better than it really is, and can cause our model to miss turning points. Hence, if you’re model is predicting an increase in sales and you, in actuality, see sales plunge, it may be due to autocorrelation.

What Does Autocorrelation Look Like?

Autocorrelation can take on two types: positive or negative. In positive autocorrelation, consecutive errors usually have the same sign: positive residuals are almost always followed by positive residuals, while negative residuals are almost always followed by negative residuals. In negative autocorrelation, consecutive errors typically have opposite signs: positive residuals are almost always followed by negative residuals and vice versa.

In addition, there are different orders of autocorrelation. The simplest, most common kind of autocorrelation, first-order autocorrelation, occurs when the consecutive errors are correlated. Second-order autocorrelation occurs when error terms two periods apart are correlated, and so forth. Here, we will concentrate solely on first-order autocorrelation.

You will see a visual depiction of positive autocorrelation later in this post.

What Causes Autocorrelation?

The two main culprits for autocorrelation are sluggishness in the business cycle (also known as inertia) and omitted variables from the model. At various turning points in a time series, inertia is very common. At the time when a time series turns upward (downward), its observations build (lose) momentum, and continue going up (down) until the series reaches its peak (trough). As a result, successive observations and the error terms associated with them depend on each other.

Another example of inertia happens when forecasting a time series where the same observations can be in multiple successive periods. For example, I once developed a model to forecast enrollment for a community college, and found autocorrelation to be present in my initial model. This happened because many of the students enrolled during the spring term were also enrolled in the previous fall term. As a result, I needed to correct for that.

The other main cause of autocorrelation is omitted variables from the model. When an important independent variable is omitted from a model, its effect on the dependent variable becomes part of the error term. Hence, if the omitted variable has a positive correlation with the dependent variable, it is likely to cause error terms that are positively correlated.

How Do We Detect Autocorrelation?

To illustrate how we go about detecting autocorrelation, let’s first start with a data set. I have pulled the average hourly wages of textile and apparel workers for the 18 months from January 1986 through June 1987. The original source was the Survey of Current Business, September issues from 1986 and 1987, but this data set was 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

Now, let’s run a simple regression model, using time period t as the independent variable and Wage as the dependent variable. Using the data set above, we derive the following model:

Ŷ = 5.7709 + 0.0095t

Examine the Model Output

Notice also the following model diagnostic statistics:

R2=

0.728

Variable

Coefficient

t-ratio

Intercept

5.7709

367.62

t

0.0095

6.55

 

You can see that the R2 is a high number, with changes in t explaining nearly three-quarters the variation in average hourly wage. Note also the t-ratios for both the intercept and the parameter estimate for t. Both are very high. Recall that a high R2 and high t-ratios are symptoms of autocorrelation.

Visually Inspect Residuals

Just because a model has a high R2 and parameters with high t-ratios doesn’t mean autocorrelation is present. More work must be done to detect autocorrelation. Another way to check for autocorrelation is to visually inspect the residuals. The best way to do this is through plotting the average hourly wage predicted by the model against the actual average hourly wage, as Middleton has done:

Notice the green line representing the Predicted Wage. It is a straight, upward line. This is to be expected, since the independent variable is sequential and shows an increasing trend. The red line depicts the actual wage in the time series. Notice that the model’s forecast is higher than actual for months 5 through 8, and for months 17 and 18. The model also underpredicts for months 12 through 16. This clearly illustrates the presence of positive, first-order autocorrelation.

The Durbin-Watson Statistic

Examining the model components and visually inspecting the residuals are intuitive, but not definitive ways to diagnose autocorrelation. To really be sure if autocorrelation exists, we must compute the Durbin-Watson statistic, often denoted as d.

In our June 24 Forecast Friday post, we demonstrated how to calculate the Durbin-Watson statistic. The actual formula is:

That is, beginning with the error term for the second observation, we subtract the immediate previous error term from it; then we square the difference. We do this for each observation from the second one onward. Then we sum all of those squared differences together. Next, we square the error terms for each observation, and sum those together. Then we divide the sum of squared differences by the sum of squared error terms, to get our Durbin-Watson statistic.

For our example, we have the following:

t

Error

Squared Error

et-et-1

Squared Difference

1

0.0396

0.0016

     

2

0.0001

0.0000

(0.0395) 0.0016

3

0.0006

0.0000

0.0005 0.0000

4

0.0011

0.0000

0.0005 0.0000

5

(0.0384)

0.0015

(0.0395) 0.0016

6

(0.0379)

0.0014

0.0005 0.0000

7

(0.0474)

0.0022

(0.0095) 0.0001

8

(0.0169)

0.0003

0.0305 0.0009

9

0.0536

0.0029

0.0705 0.0050

10

0.0041

0.0000

(0.0495) 0.0024

11

(0.0054)

0.0000

(0.0095) 0.0001

12

0.0152

0.0002

0.0205 0.0004

13

0.0457

0.0021

0.0305 0.0009

14

0.0262

0.0007

(0.0195) 0.0004

15

0.0167

0.0003

(0.0095) 0.0001

16

0.0172

0.0003

0.0005 0.0000

17

(0.0423)

0.0018

(0.0595) 0.0035

18

(0.0318)

0.0010

0.0105 0.0001
  

Sum:

0.0163

  

0.0171

 

To obtain our Durbin-Watson statistic, we plug our sums into the formula:

= 1.050

What Does the Durbin-Watson Statistic Tell Us?

Our Durbin-Watson statistic is 1.050. What does that mean? The Durbin-Watson statistic is interpreted as follows:

  • If d is close to zero (0), then positive autocorrelation is probably present;
  • If d is close to two (2), then the model is likely free of autocorrelation; and
  • If d is close to four (4), then negative autocorrelation is probably present.

As we saw from our visual examination of the residuals, we appear to have positive autocorrelation, and the fact that our Durbin-Watson statistic is about halfway between zero and two suggests the presence of positive autocorrelation.

Next Forecast Friday Topic: Correcting Autocorrelation

Today we went through the process of understanding the causes and effect of autocorrelation, and how to suspect and detect its presence. Next week, we will discuss how to correct for autocorrelation and eliminate it so that we can have more efficient parameter estimates.

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