## Time Series – Interventions and Contribution

- 13/05/2016
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**Published In**

- Big Data
- Analytics
- Business Intelligence

This article illustrates principles of an analysis of the President George W. Bush’s job approval from January 2001 through Sep 2004 with disposable income excluded from the statistical model. To see a version complete with code and its description, visit bicorner.com.

Presidents with a job approval rating of less than 50 percent are unlikely to be re-elected. During June, Bush’s job approval rating averaged 47 percent in five major polls.

We note immediately the unusual, roller-coaster characteristic of the Bush Presidency, divided into several distinct phases (See Figure 1). A first phase, lasting through September 10, 2001, resembles the course of many other presidencies: approval begins to decline soon after the inauguration, although it may respond to short-term events and the presidents’ policy successes and failures, especially on economic issues. The second phase began on September 11, 2001. In the aftermath of the attacks in New York and Washington, Bush’s approval surged to the highest level ever recorded, and well into 2002, it remained above the average for his entire term. This was the largest and longest “rally” in presidential approval that has ever been recorded. A third phase of the Bush Presidency began with the war in Iraq when his slowly declining approval from September 11th surged once again after the war began (although not to the levels of fall 2001). After the end of major hostilities on May 1, 2003, approval began a long decline to the 47 percent level cited immediately above.

Since we are only looking at job approval rating, we will exclude the third phase, for we do not know how much of the decline in the post-Iraq war phase was due to the unpopularity of the war and occupation, and in particular to the casualties that were been suffered. This is quite different than an emergency response to an attack on our country on 9/11.

The President’s Job Approval Rating takes account of citizen policy evaluations beyond the single issue of the War with Iraq. We consider here the external factors: (1) Executive Response to the World Trade Center terrorist attack, (2) disposable personal income growth per capita (RDPI), and (3) a dummy variable reflecting a change on September 9, 2001.

The data can be downloaded in Excel format (then saves as .CSV) from http://www.owlnet.rice.edu/~stoll/bushpop/fpc.xls.

**Intervention**

Suppose we wanted to analyze the impact of a change in policy. Policy changes are often used to intervene in some state of distress, such as recession, natural disaster, foreign policy, and so on. In this time series analysis, the intervention came in the form of President Bush’s response to the World Trade Center attacks on September 11, 2001. A second, but one with less impact, was the President’s decision to invade Iraq in 2003. We can see both of these in the time series plot below.

The time series model used in this analysis was an ARIMA(0,1,0) model. As discussed in a previous post, ARIMA interactively applies Auto-Regressive and Moving Average component, along with a dependent lag. For this model, however, only a dependent lag is used, for instance, ARIMA(AR=0, Dep. Lag=1, MA=0). The predictors used in the model included the RDPI as previously mentioned and a dummy variable for the event/date September 11, 2001. The intervention in this time series in the Executive Response to the terrorist attacks on the World Trade Center. We are not considering the second intervention, the Executive decision to invade Iraq.

To help determine the actual value of the external predictor variable WTC

y(t) = β(t) + αx(t – 1) + ε(t)

where y(t) is the dependent variable, y(t – 1)is the dependent lagged variable, is α the coefficient of the dependent lag, x(t) is the independent variable with coefficient β and ε(t) is the error term.

The simplest example of a contribution chart involves one explanatory variable with on the lagged dependent variable (LDV). In this instance, the lagged dependent variable functions as a feedback loop, and the coefficient on the LDV defines the magnitude of this amplification. So, a naïve way of constructing the contribution chart is to simply plot the contemporaneous contribution of the independent variable, magnified by the geometric factor implied by the LDV.

Y(t) = [b /(1-a)] x(t)

where Y(t), b, a are estimates of their associated parameters.

This is the most straightforward approach but assumes all amplification is instantaneous, which is not the case. This naïve approach can be easily implemented in a spreadsheet, using the forecasted data generated by the model, e.g., PROC LOGISTIC in SAS Studio.

**Contribution Chart**

The contribution of WTC toward the Presidential Job Approval Ratings is shown in Figure 2. This intervention event clearly had an impact on President Bush’s job approval ratings. Though the intervention ended about 23 December 2001, the contribution chart depicts the forecasted contribution as well. So, WTC continues to have an impact after it has officially ended.

**UCM Model Parameters**

The model parameters for the independent variable (see Table 1) and the dependent lag coefficients (see Table 1), and the Level (see Table 2 for the intercept component) are required for implementing Equation 1.

Table 1. Model parameters for the independent variables

Table 2. Trend Information for Final UCM Model

**Excel Cell Formulas**

The Excel formulas representing Equation 1 are shown below.

**The Art of Modeling**

We did not include SEP11 in the model, even though it is significant since the two independent variables SEP11 and WTC are highly correlated, and the effects are captured by WTC. The contribution chart including all three independent variables reveals why RDPI is not significant as it contributes very little to the Job Approval Ratings.

**Conclusion**

A contribution chart in time series analysis is an excellent way to understand how significant predictor variables impact the dependent variable as demonstrated by WTC for the Presidential Job Approval Ratings.

- 13/05/2016
- 1078
- 0 Like

## Time Series – Interventions and Contribution

- 13/05/2016
- 1078
- 0 Like

#### Jeffrey Strickland

Predictive Analytics Consultant at Clarity Solution Group

Opinions expressed by Grroups members are their own.

#### Top Authors

This article illustrates principles of an analysis of the President George W. Bush’s job approval from January 2001 through Sep 2004 with disposable income excluded from the statistical model. To see a version complete with code and its description, visit bicorner.com.

Presidents with a job approval rating of less than 50 percent are unlikely to be re-elected. During June, Bush’s job approval rating averaged 47 percent in five major polls.

We note immediately the unusual, roller-coaster characteristic of the Bush Presidency, divided into several distinct phases (See Figure 1). A first phase, lasting through September 10, 2001, resembles the course of many other presidencies: approval begins to decline soon after the inauguration, although it may respond to short-term events and the presidents’ policy successes and failures, especially on economic issues. The second phase began on September 11, 2001. In the aftermath of the attacks in New York and Washington, Bush’s approval surged to the highest level ever recorded, and well into 2002, it remained above the average for his entire term. This was the largest and longest “rally” in presidential approval that has ever been recorded. A third phase of the Bush Presidency began with the war in Iraq when his slowly declining approval from September 11th surged once again after the war began (although not to the levels of fall 2001). After the end of major hostilities on May 1, 2003, approval began a long decline to the 47 percent level cited immediately above.

Since we are only looking at job approval rating, we will exclude the third phase, for we do not know how much of the decline in the post-Iraq war phase was due to the unpopularity of the war and occupation, and in particular to the casualties that were been suffered. This is quite different than an emergency response to an attack on our country on 9/11.

The President’s Job Approval Rating takes account of citizen policy evaluations beyond the single issue of the War with Iraq. We consider here the external factors: (1) Executive Response to the World Trade Center terrorist attack, (2) disposable personal income growth per capita (RDPI), and (3) a dummy variable reflecting a change on September 9, 2001.

The data can be downloaded in Excel format (then saves as .CSV) from http://www.owlnet.rice.edu/~stoll/bushpop/fpc.xls.

**Intervention**

Suppose we wanted to analyze the impact of a change in policy. Policy changes are often used to intervene in some state of distress, such as recession, natural disaster, foreign policy, and so on. In this time series analysis, the intervention came in the form of President Bush’s response to the World Trade Center attacks on September 11, 2001. A second, but one with less impact, was the President’s decision to invade Iraq in 2003. We can see both of these in the time series plot below.

The time series model used in this analysis was an ARIMA(0,1,0) model. As discussed in a previous post, ARIMA interactively applies Auto-Regressive and Moving Average component, along with a dependent lag. For this model, however, only a dependent lag is used, for instance, ARIMA(AR=0, Dep. Lag=1, MA=0). The predictors used in the model included the RDPI as previously mentioned and a dummy variable for the event/date September 11, 2001. The intervention in this time series in the Executive Response to the terrorist attacks on the World Trade Center. We are not considering the second intervention, the Executive decision to invade Iraq.

To help determine the actual value of the external predictor variable WTC

y(t) = β(t) + αx(t – 1) + ε(t)

where y(t) is the dependent variable, y(t – 1)is the dependent lagged variable, is α the coefficient of the dependent lag, x(t) is the independent variable with coefficient β and ε(t) is the error term.

The simplest example of a contribution chart involves one explanatory variable with on the lagged dependent variable (LDV). In this instance, the lagged dependent variable functions as a feedback loop, and the coefficient on the LDV defines the magnitude of this amplification. So, a naïve way of constructing the contribution chart is to simply plot the contemporaneous contribution of the independent variable, magnified by the geometric factor implied by the LDV.

Y(t) = [b /(1-a)] x(t)

where Y(t), b, a are estimates of their associated parameters.

This is the most straightforward approach but assumes all amplification is instantaneous, which is not the case. This naïve approach can be easily implemented in a spreadsheet, using the forecasted data generated by the model, e.g., PROC LOGISTIC in SAS Studio.

**Contribution Chart**

The contribution of WTC toward the Presidential Job Approval Ratings is shown in Figure 2. This intervention event clearly had an impact on President Bush’s job approval ratings. Though the intervention ended about 23 December 2001, the contribution chart depicts the forecasted contribution as well. So, WTC continues to have an impact after it has officially ended.

**UCM Model Parameters**

The model parameters for the independent variable (see Table 1) and the dependent lag coefficients (see Table 1), and the Level (see Table 2 for the intercept component) are required for implementing Equation 1.

Table 1. Model parameters for the independent variables

Table 2. Trend Information for Final UCM Model

**Excel Cell Formulas**

The Excel formulas representing Equation 1 are shown below.

**The Art of Modeling**

We did not include SEP11 in the model, even though it is significant since the two independent variables SEP11 and WTC are highly correlated, and the effects are captured by WTC. The contribution chart including all three independent variables reveals why RDPI is not significant as it contributes very little to the Job Approval Ratings.

**Conclusion**

A contribution chart in time series analysis is an excellent way to understand how significant predictor variables impact the dependent variable as demonstrated by WTC for the Presidential Job Approval Ratings.

- 13/05/2016
- 1078
- 0 Like

## Jeffrey Strickland

Predictive Analytics Consultant at Clarity Solution Group

Opinions expressed by Grroups members are their own.