Question

In the National Football League, the philosophy for winning (rushing, passing, defense) seems to go through cycles. Consider a time series of the average number of rushing yards in the NFL per regular season from 1980 to 2008. Make a time series plot. Is there evidence that the average rushing yards is trending in one direction? Describe the general movement of the series. Fit a first order autoregressive model [AR(1)] using y(t) as the response variable and y (t-1) as the input variable. Record the regression equation. Based on the AR(1) model, forecast the average number of rushing yards in the NFL for the 2009 regular season. Calculate the exponential smoothing models using Excel damping factors 0.8 and 0.2 For each of the exponential smoothing models forecast the average number of rushing yards in the NFL for the 2009 season. Calculate a moving average model using k=5 (Excel interval). Forecast the average number of rushing yards in the NFL for the 2009 season.

Answer 1

1.Required plot in excel.

140 120 100 80 60 40 20 1975 1980 1985 1990 1995 2000 2005 2010 2015 Year yLinear (y)

yes the average rushing yard is trending in one derection and is going downwards which can be seen from the linear dotted lines.

2. We find AR[1] using R

> arima(x1,or der-c(1,0,0)) Call arima(x x1, orderc(1, 0, 0)) Coefficients: arl intercept 0.7036 116. 3078 2.9578 s.е. 0.1337 sigmaA2 estimated as 25.39: 1og > y<-ar i ma(x1, order=c (1,0,0)) > forecast (y) likelihood-88. 39 aic182.78 Point Forecast L080 Hİ 80 L095 Hİ95 115.1062 108. 6482 121.5641 105. 2296 124. 9828 30

x1 is the series having values.

x1<-c(127.5,130.1,117.8,129.7,123.9,124.9,118.7,123.9,121.4,115.3,113.9,107.7,110.5,110,104.3,108.1,109,113,112.7,106.5,112.6,106.5,112.6,117.9,116.6,112.5,117.3,110.9,114.6)

The required AR[1] model is Y(t)=116.3078+0.7036Y(t-1)

3. Forecasted value for 2009 is 115.12

4. Exponential Smoothing

NFL Data alpha .8 alpha 2 1 1980 127.5 #N/A #N/A 2 1981 130.1 127.5127.5 З 1982 117.8 129.58 128.02 4 1983 129.7 120.156 125.976 1984 123.9 127.7912 126.7208 6 1985 124.9 124.6782 126.1566 7 1986 118.7 124.8556 125.9053 8 1987 123.9 119.9311 124.4642 91988 121.4 123.1062 124.3514 10 1989 115.3 121.7412 123.7611 11 1990113.9 116.5882 122.0689 1991107.7 114.4376 120.4351 13 1992110.5 109.0475 117.8881 110 110.2095 116.4105 15 1994 104.3 110.0419 115.1284 16 1995 108.1 105.4484 112.9627 17 1996109 107.5697 111.9902 113 108.7139 111.3921 19 1998112.7 112.1428 111.7137 20 1999106.5 112.5886 111.911 21 2000 112.6 107.7177 110.8288 22 2001106.5 111.6235 111.183 23 2002 112.6 107.5247 110.2464 24 2003 117.9 111.5849 110.7171 25 2004 116.6 116.637 112.1537 26 2005 112.5 116.6074 113.043 27 2006117.3 113.3215 112.9344 28 2007 110.9 116.5043 113.8075 29 2008 114.6 112.0209 113.226 114.0842 113.5008 6 9841291295 14 1993 18 1997 30 2009

Exp Smoothing 135 130 125 120 115 110 105 100 1975 1980 1985 1990 1995 2000 2005 2010 2015

the forecasted values are114.08 and 113.5 for damping factor 0.2 and 0.8 respectively.