Exam 23: Time-Series Analysis and Forecasting

Quarterly sales revenue (in $million) for a particular company has been modelled using linear regression with indicator variables: Y = 132 + 2Q1 + 3Q2 - 5Q4 + 2t Where t is time in quarters, with origin March 2006 and Q1, Q2 and Q4 are the indicator variables for March, June and December quarters, respectively. Describe the trend and seasonal effects.

The trend line $=125+2 t$ and seasonal indexes shown below were computed from 10 years of quarterly data. Forecast the values for the next four quarters. Quarter S 1 0.6 2 1.3 3 1.6 4 0.5

The following linear trend was estimated using a time series regression with the origin in the year 2000. ŷ = 76.80 + 3.14t. Which of the following is the forecast for the year 2013?

A time series regression equation measuring the number of surfboards sold by a surfboard manufacturing company in Australia is given below: Y = 35 + 4Q₁ + 0.5Q₃ + 8Q₄ + 3t With t in quarters and the origin is December 2010 and Q₁ is the indicator variable for March, Q₃ is the indicator variable for September and Q₄ is the indicator variable for December. Which of the following statements is correct regarding the trend component?

A company selling swimming goggles wants to analyze its Australian sales figures. Time series forecasting with regression was used to generate Excel output to estimate trend and seasonal effects of the time series of Swimming goggle sales (in thousands of dollars) where the origin is the March Quarter 2000 and Q1 denotes sales in the March quarter, Q3 denotes sales in the September quarter and Q4 denotes sales in the December quarter. SUMMARV OUTPUT Regression Stotitics Multiple R 0.9460 R Square 0.8950 Adjusted RSquare 0.8864 Standard Error 3.7394 Observations 54 $\text { ANOVA }$ Sgnificance df S MS F F Regression 4 5837.596003 1459.4 104.3701 2.41949-23 Residual 49 685.1632564 13.9829 Total 53 6522.759259 Standard Upper Coefficients Error t Stat p-value Lower 95\% 95\% Intercept 3.0588 1.3331 2.2944 0.0261 0.3797 5.7378 0.2518 0.0327 7.7052 0.0000 0.1861 0.3175 1 12.4604 1.3897 8.9664 0.0000 9.6677 15.2530 3 1.1458 1.4721 0.7784 0.4401 -1.8124 4.1041 4 23.9121 1.4403 16.6025 0.0000 21.0177 26.8064 (a) Forecast swimming goggle sales for all four quarters of 2016. (b) Are these good forecasts? Explain. (c) Separate the difference in your forecasts for June 2016 and December 2016 between seasonal and trend.

The most commonly used measures of forecast accuracy are the:

The quarterly earnings of a large microcomputer company have been recorded for the years 1993-1996. These data (in millions of dollars) are shown in the accompanying table. Year Quarter 1993 1994 1995 1996 1 60 65 68 74 2 75 83 85 90 3 93 98 102 106 4 62 69 71 75 Using an appropriate moving average, measure the quarterly variation by computing the seasonal (quarterly) indexes.

A company selling swimming goggles wants to analyze its Australian sales figures. Time series forecasting with regression was used to generate Excel output to estimate trend and seasonal effects of the time series of Swimming goggle sales (in thousands of dollars) where the origin is the March Quarter 2000 and Q1 denotes sales in the March quarter, Q3 denotes sales in the September quarter and Q4 denotes sales in the December quarter. SUMMARV OUTPUT Regression Stotitics Multiple R 0.9460 R Square 0.8950 Adjusted RSquare 0.8864 Standard Error 3.7394 Observations 54 $\text { ANOVA }$ Sgnificance df S MS F F Regression 4 5837.596003 1459.4 104.3701 2.41949-23 Residual 49 685.1632564 13.9829 Total 53 6522.759259 Standard Upper Coefficients Error t Stat p-value Lower 95\% 95\% Intercept 3.0588 1.3331 2.2944 0.0261 0.3797 5.7378 0.2518 0.0327 7.7052 0.0000 0.1861 0.3175 1 12.4604 1.3897 8.9664 0.0000 9.6677 15.2530 3 1.1458 1.4721 0.7784 0.4401 -1.8124 4.1041 4 23.9121 1.4403 16.6025 0.0000 21.0177 26.8064 (a) Using p-values test the significance of the independent variables. (b) Test the significance of the overall regression equation.

The linear model for long-term trend is $y = \beta _ { 0 } + \beta _ { 1 } t + \varepsilon$ , where t is the time period. The trend is indicated by:

The effect that business recessions and prosperity have on time-series values is an example of the disaster component of a time series.

In general, it is easy to identify the trend component of a time series by using:

Weekly toy sales (in $1000s) in a department store for the past three months are shown below. Month Week Sales 1 1 14 2 22 3 20 4 16 2 1 18 2 20 3 24 4 20 3 1 22 2 26 3 24 4 18 Compute the four-week centred moving averages.

A time series for the years 1990-1995 is shown below. Year 1990 125 1991 115 1992 120 1993 126 1994 140 1995 122 a. Develop forecasts for the years 1996-1998, with the following smoothing constant values: w = 0.2, w = 0.5 and w = 0.6. b. Compare each of the three sets of forecasts above with the actual values for 1996-1998 given in the following table, and compute the MAD for each model. Which model is best? Year 1996 130 1997 125 1998 135

Which of the following best describes a time series?

a. The seasonally adjusted US quarterly Industrial Production Index from the first quarter of 2001 to the fourth quarter of 2005 (y_t, 2002 = 100) is shown in the table below. Would the linear or quadratic model fit better? Time period Mar-01 136.7 Jun-01 124.1 Sep-01 120.5 Dec-01 117.4 Mar-02 101.1 Jun-02 102.5 Sep-02 98.5 Dec-02 97.9 Mar-03 94.0 Jun-03 86.7 Sep-03 89.8 Dec-03 92.3 Mar-04 95.9 Jun-04 89.6 Sep-04 86.3 Dec-04 84.5 Mar-05 88.7 Jun-05 109.9 Sep-05 100.9 Dec-05 108.4 b. Use Excel and the regression technique to calculate the linear trend line and the quadratic trend line. Which model fits better?

The number of pairs of sunglasses sold each quarter in a beachside drugstore were recorded for the years 2007-2010. These data are shown in the following table. Year Quarter 2007 2008 2009 2010 1 82 84 85 90 2 72 71 70 74 3 65 66 67 71 4 53 54 56 58 a. Develop a regression model, using indicator variables to represent quarters. b. Forecast the quarterly earnings for the years 2011 and 2012.

A trend is one of the four different components of a time series. It is a long-term, relatively smooth pattern or direction exhibited by a series, and its duration is more than one year.

If we wanted to measure the seasonal variations on stock market performance by quarter, we would need:

To calculate the random component of a time series, ignoring the cyclical component, it would be the difference between an actual observation and the predicted value using a regression model with indicator variables for the seasonal component and time as the trend component.

The following trend line was calculated from quarterly data for 2006-2010: ŷ = 2.35 + 0.12t, where t = 1 for the first quarter of 2006. The trend value for the third quarter of the year 2011 is: