Chapter 10 – Basic Regression Analysis with Time Series Data

Chapter 10 – Basic Regression Analysis with Time Series Data

What is Time Series Data and why is it Different?

• There is a time ordering of the data• The past can affect the future, but the future

cannot affect the past.

Example: National population from 1900 to 2006 (data set NATPOP)

What is Time Series Data and why is it Different?

• Random nature of times series data• Formally, the process that generates time

series data is called a stochastic or time series process

What is Time Series Data and why is it Different?

• Random nature of times series data• Random sample from a population vs.

random sample of time series data

Examples of Time Series Data: Uncorrelated data, constant process model

Examples of Time Series Data: Autocorrelated data

Examples of Time Series Data: Trend

Examples of Time Series Data: Cyclic or seasonal data

Examples of Time Series Data: Nonstationary data

Examples of Time Series Data: A mixture of patterns

Cyclic patterns of different magnitudes

Atypical events

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Atypical events

Famous Time Series Expert –Yogi Berra

The future ain’t what it used to be.

Famous Time Series Expert –Yogi Berra

You can observe a lot just by watching.

The basic graphical display for time series data is the time series plot which is just a graph of the observations vs. time periods.

Time Series Plot Example

Open TRAFFIC2 data set and make time series plot of year vs. statewide total accidents (totacc)

In Minitab need to choose series and time stamp

Time Series Plot Example

Time series plots

Notice that the histograms look very similar even though the time series behavior is very different

Histogram of totacc

When there are two or more variables of interest, scatter plots can be useful

Forecasting

It is difficult to make predictions, especially about the future. – Neils Bohr

Forecasting

Forecasting is useful in many fields:

Business and industryEconomicsFinanceEnvironmental sciencesSocial sciencesPolitical sciences

Data Analysis Process:

1. Problem definition2. Data collection3. Data analysis4. Model selection and fitting5. Model validation6. Model deployment7. Monitoring forecasting model

performance

Time Series Example – Data Set FERTIL3

gfr – number of children born to every 1,000 women of childbearing age from 1913 to 1984.

Make a time series plot of gfr

Time Series Example – Data Set FERTIL3

Time Series Example – Data Set FERTIL3

pe – average real dollar value of the personal tax exemption from 1913 to 1984.

Make a time series plot of pe

Time Series Example – Data Set FERTIL3

Time Series Example – Data Set FERTIL3

We want to predict gfr.

Lets try this model:

Time Series Example – Data Set FERTIL3

We want to predict gfr.

Notation for time series model slightly different:

Time Series Example – Data Set FERTIL3

The regression equation isgfr = 96.3 - 0.0071 pe

Predictor Coef SE Coef T PConstant 96.344 4.305 22.38 0.000pe -0.00710 0.03592 -0.20 0.844

S = 19.9400 R-Sq = 0.1% R-Sq(adj) = 0.0%

Time Series Example – Data Set FERTIL3

residual plots

Time Series Example – Data Set FERTIL3

residual plots

Time Series Example – Data Set FERTIL3

residual plots

Time Series Example – Data Set FERTIL3

residual plots

Time Series Example – Data Set FERTIL3

This model suffers from misspecification.

Time Series Example – Data Set FERTIL3Scatter plot of gfr vs. pe

Time Series Example – Data Set FERTIL3What could affect general fertility rate in the U.S.? Many things!How about these two:• World War II• Availability of the birth control pill

Time Series Example – Data Set FERTIL3ww2 is a dummy variable• 1 if year is 1941 through 1945• 0 otherwisepill is a dummy variable • 1 if year is 1963 or greater• 0 otherwise

Time Series Example – Data Set FERTIL3

Fit this model with two dummy variables.

Time Series Example – Data Set FERTIL3

gfr = 98.7 + 0.0825 pe - 24.2 ww2 - 31.6 pill

Predictor Coef SE Coef T PConstant 98.682 3.208 30.76 0.000pe 0.08254 0.02965 2.78 0.007ww2 -24.238 7.458 -3.25 0.002pill -31.594 4.081 -7.74 0.000

S = 14.6851 R-Sq = 47.3% R-Sq(adj) = 45.0%

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

gfr = 98.7 + 0.0825 pe - 24.2 ww2 - 31.6 pillww2 is a dummy variable• 1 if year is 1941 through 1945• 0 otherwise

gfr = 98.7 + 0.0825 pe - 24.2 ww2 - 31.6 pillpill is a dummy variable • 1 if year is 1963 or greater• 0 otherwise

Time Series Example – Data Set FERTIL3

gfr = 98.7 + 0.0825 pe - 24.2 ww2 - 31.6 pill

Summary:• Adding pe and ww2 improves model significantly• Model gives insight into historical variables that

affect gfr• Model may not be very useful for future predictions

Time Series Example – Data Set FERTIL3

Modeling with lags

Economic theory implies that there might be a lag effect on gfr (general fertility rate) from pe (tax value of having a child)

Time Series Example – Data Set FERTIL3

Modeling with lags

Examine variables pe_1, pe_2, pe_3, and pe_4

Time Series Example – Data Set FERTIL3

Making lags in Minitab is easy. Go to Stat > Time Series > Lag.

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

Residual plots:

Time Series Example – Data Set FERTIL3

gfr = 95.9 + 0.073 pe - 0.006 pe_1 + 0.034 pe_2 - 22.1 ww2 - 31.3 pill

Predictor Coef SE Coef T PConstant 95.870 3.282 29.21 0.000pe 0.0727 0.1255 0.58 0.565pe_1 -0.0058 0.1557 -0.04 0.970pe_2 0.0338 0.1263 0.27 0.790ww2 -22.13 10.73 -2.06 0.043pill -31.305 3.982 -7.86 0.000

S = 14.2701 R-Sq = 49.9% R-Sq(adj) = 45.9%

Time Series Example – Data Set FERTIL3

Personally, I think adding the two lags to this model over complicates the model for little gain. I recommend against inclusion of the two lags.