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Exam 23: Time-Series Analysis and Forecasting

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Question 61

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A time series can consist of four different components: long-term trend, cyclical variation, seasonal variation, and random variation.

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Question 62

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Regression analysis with t = 1 to 40 was used to develop the following equation: $=1500+5 t+1.5 Q_{1}+1.8 Q_{2}-3.0 Q_{3}$ , where: $Q _ { i }$ = 1, if quarter i (i = 1, 2, 3) = 0, otherwise. Forecast the next four quarters.

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Question 63

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The time-series component that reflects a wavelike pattern describing a long-term trend that is generally apparent over a number of years is called seasonal.

The total number of overtime hours (in 1000s) worked in a large steel mill was recorded for 16 quarters, as shown below. $\begin{array} { | c | c | c | } \hline \text { Year } & \text { Quarter } & \text { Overtime hours } \\\hline 2007 & 1 & 20 \\& 2 & 30 \\& 3 & 28 \\& 4 & 20 \\2008 & 1 & 24 \\& 2 & 34 \\& 3 & 28 \\& 4 & 21 \\2009 & 1 & 28 \\& 2 & 38 \\& 3 & 31 \\& 4 & 26 \\2010 & 1 & 30 \\& 2 & 41 \\& 3 & 35 \\& 4 & 28 \\\hline\end{array}$ a. Use the regression technique to calculate the linear trend line. b. Calculate the seasonal indexes based on the regression trend line in part (a). c. What do the seasonal indexes tell us?

Regression analysis with t = 1 to 80 was used to develop the following forecast equation: $\hat{y}$ _t = 135 + 4.8t -1.3Q₁ -1.7Q₂ + 1.5Q₃ where: Q_i = 1, if quarter i (i = 1, 2, 3) = 0, otherwise. Forecast the next four values.

The term 'seasonal variation' may refer to the four traditional seasons, or to systematic patterns that occur during a quarter, a week, or even one day, but within 12 months.

The result of a quadratic model fit to time-series data was $=8.5-0.25 t+2.5 t^{2}$ , where t = 1 for 1994. The forecast value for 2001 is 129.25.

Which of the following best describes what may be used when measuring the seasonal and random variation of a time series with no cyclical effect?

Of the four different components of a time series, cyclical variation is the one most likely to exhibit the long-term direction of the data.

a. Apply exponential smoothing with w = 0.1 and w = 0.8 to help detect the components of the following time series. $\begin{array} { c c } \text { Period } t & y \mathrm { t } \\\hline 1 & 40 \\2 & 45 \\3 & 44 \\4 & 47 \\5 & 48 \\6 & 50 \\7 & 52 \\8 & 51 \\9 & 48 \\10 & 47\end{array}$ b. Draw the time series and the two sets of exponentially smoothed values. Does there appear to be a trend component in the time series?

The mean absolute deviation (MAD) and the sum of squares for forecast error (SSE) are the most commonly used measures of forecast accuracy. The model that forecasts the data best will usually have the:

Which of the following are examples of seasons when measuring the seasonal component of a time series?

A company selling swimming goggles wants to analyze the company's Australian sales figures. Time series forecasting with regression was used to generate Excel output to estimate trend of the time series of Swimming goggle sales (in thousands of dollars) where the origin is the March Quarter 2000. $\begin{array}{l}\text { SUMMARV OUTPUT }\\\hline\text { Regression Stotitics }\\\begin{array}{lc}\hline \text { Multiple R } & 0.37281 \\\text { R Square } & 0.13899 \\\text { Adjusted R } & \\\text { Square } & 0.12243 \\\text { StandardError } & 10.3925 \\\text { Observations } & 54\\\hline\end{array}\end{array}$ $\text { ANOVA }$ $\begin{array}{lccccc} \hline& & & & & \text { Sgnificance } \\& d f & S & M S & F & F \\\hline \text { Regression } & 1 & 906.5867925 & 906.59 & 8.39406 & 0.005497292 \\\text { Residual } & 52 & 5616.172467 & 108 & & \\\text { Total } & 53 & 6522.759259 & & & \\\hline\end{array}$ $\begin{array}{lcrrrrr}\hline&& \text { Standard } & & && \text { Upper } \\&\text { Coefficients } & \text { Error } & \text { t Stat } & \text {p-value } & \text { Lower } 95 \%& 95 \%\\\hline\text { Intercept } & 12.237 & 2.789633876 & 4.3866 & 5.6 \mathrm{E}-05 & 6.639227133 & 17.8348469 \\\mathrm{t} & 0.26289 & 0.090738795 & 2.8973 & 0.0055 & 0.080812368 & 0.4449738 \\\hline\end{array}$ (a) Forecast goggles sales for each quarter of 2016. (b) Are these good predictions? Explain.

The actual and forecast values of a time series are shown below. $\begin{array} { | c c | } \hline \text { Actual values } y _ { t } & \text { Forecast values } F _ { t } \\\hline 135 & 140 \\162 & 165 \\155 & 150 \\182 & 191 \\174 & 168 \\194 & 190 \\233 & 220 \\280 & 240 \\\hline\end{array}$ a. Calculate the mean absolute deviation (MAD). b. Calculate the sum of squares for forecast error (SSE).

To calculate the five-period moving average of a time series for a given time period, we average the value in that time period and the values in the four preceding periods.

The time-series component that reflects a long-term, relatively smooth pattern or direction exhibited by a time series over a long time period is called trend.

In determining weekly seasonal indexes for natural gas consumption, the sum of the 52 means for gas consumption as a percentage of the moving average is 5195. To get the seasonal indexes, each monthly mean is to be multiplied by (5200 / 5195).

The stock market has a 5-day working week. If we wanted to measure the impact of the day of the week on stock market performance, we would need:

The table below shows the number of pizzas sold daily during a four-week period at King Pizza in Melbourne. $\begin{array}{l}\quad\quad\quad\quad\quad\quad\quad\quad\text { Week }\\\begin{array} { | l | r r r c | } \hline\text { Day } & 1 & 2 & 3 & 4 \\\hline\text { Sunday } & 253 & 234 & 248 & 232 \\\text { Monday } & 98 & 93 & 99 & 104 \\\text { Tuesday } & 106 & 88 & 87 & 115 \\\text { Wednesday } & 119 & 134 & 113 & 102 \\\text { Thursday } & 138 & 123 & 130 & 118 \\\text { Friday } & 201 & 215 & 218 & 205 \\\text { Saturday } & 327 & 399 & 415 & 390 \\\hline\end{array}\end{array}$ a. Calculate the seasonal (daily) indexes, using a seven-day moving average. b. Use regression analysis to find the linear trend line. c. Calculate the seasonal (daily) indexes, using the trend line developed in (b).

Which of the following methods is appropriate for forecasting a time series when the trend, cyclical and seasonal components of the series are not significant?