Question

#1.8 The MSE-better forecast for the foregoing Moving Average models is #1.9 The MSE-better forecast for the foregoing Exponential Smoothing models is #1.10 The MSE-best model among the foregoing Moving Average, Exponential Smoothing, and Linear Regression models is because Year Period (t)Enrollment (1000s) 2001 2002 2003 2004 2005 2006 2007 2008 2009 6.5 8.1 8.4 10.2 12.5 13.3 13.7 17.2 18.1 4

Answer 1

1.8) MSE better estimator

To get the pin pointed MSE (mean squared estimate) the interval in the moving average should be taken as a lower value.

The smaller the interval, the closer the moving averages are to the actual data points.

Thus we chose an interval of 3 to get the better foregoing moving average model

I have also derived a 4 interval moving average(MA) model to illustrate the siginificant shift in MSE as we move from 4 interval to 3 interval MA model.(foregoing)

Moving Average Model (1) Simple MA models Forecastvs. Actual 3-month MA 4-month MA Enrollment 20 18 16 14 12 8 10 E8 period month(00)forecast error ABS(error)forecasteror ABS(error) 6.5 8.1 8.4 10.2 12.5 13.3 13.7 17.2 18.1 2001 2002 2003 7.67 8.90 10.37 12.00 13.17 14.73 2.53 3.60 2.93 1.70 4.03 3.37 2.53 3.60 2.93 1.70 4.03 3.37 8.30 9.80 11.10 12.43 14.18 4.20 3.50 2.60 4.78 3.93 2005 2007 2008 2009 4.20 3.50 2.60 4.78 3.93 Enrollment3-month MA forecast-4-month MA forecast (000) 3.03 BIAS 3.03 MAD 9.75 MSE 3.80 BIAS 3.80 MAD 14.97 MSE

1.9) Similarly we may find the MSE for the foregoing exponential model as well.

We will find the MSE by both the simple foregoing exponential model and then the holts' model as well.

I have made an excel sheet to calculate the same for the two exponential models.

Simple Exponential Smoothing Forecastvs. Actual Alpha nroll 0.3 Alpha -0.599 20 nt SquaredPercent period month co00) forecast error ABS(errforecast error ABS(error)ErrorError 2001 2002 2003 6.5 6.50 6.50 6.98 7.41 8.24 9.52 15 6.50 7.46 8.02 9.33 1.60 1.42 2.79 4.26 3.78 2007 13.710.65 3.05 5.63 4.84 3.42 MAD 1.60 1.42 2.79 4.26 3.78 3.05 5.63 4.84 13.65 MS 1.60 0.94 2.18 3.17 11.23 2.07 123 13.21 3.99 2.50 2.21 1.60 0.94 2.18 3.17 2.07 123 3.99 2.50 2.21 MAD 2.56 0.89 4.74 10.07 4.30 1.52 15.95 6.26 78 0.20 0.11 0.21 0.25 0.16 0.09 0.23 0.14 0.17 MAPE 8.4 10.2 12.5 13.3 10 2005 2006 12.47 2008 17.2 11.57 13.26 3.42 15.60 EnrollmentFcst (alpha-0.3) Fcst (alpha-0.5) 000)

Holt's model

4 3E 2 8 00000000-0 5746109 2 101210201 M 101210201 B 50 54 34 92 24 6789 12 13 14 17 30 97 27 21 77 83 09 667791211 0 t 00 s1425372 68802337

The linear regression model (using scatter plot and the trend line)

Trend Model (Fit a line to the time series data) Forecastvs. Actual Squared Percent period month (ooo) forecast error ABS(error) Error Error 0.05 0.06 0.08 2001 2003 2005 2007 6.5 6.17 7.63 9.09 10.54 12.00 0.50 13.46 -0.1 14.91 0.33 0.47 0.69 0.34 0.50 0.16 1.21 0.83 0.47 -0.69 -0.34 0.11 0.22 0.47 8.4 y 1.4587x +4.7167 12.5 13.3 13.7 17.2 0.25 0.02 1.47 0.69 0.04 0.01 1.21 16.37 0.83 Enrollment +forecast (000) 2008 0.05 17.83 0.00 0.53 MAD 8 MAPE Method 1: Use Excel's function 4.716667 b1.456667 Use Chart Trendline from the scatter plot

Now we have got the MSE from all the models .

Lets compare it

1 Model Comparison and Selection 3 Models 4 Trend (linear reg) 5 Moving Avg 6 Exponential Smoothing 5.78 (using Solver to find the best alpha 0.599 which minimizes MAD) 7 Holt's MSE 0.38 9.75 (best weighted moving average with n-3) 2.47 (with alpha0.5 and beta-0.3)

Clearly the linear regression model gives us the best fit since the MSE is least.