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Exam 23: Time-Series Analysis and Forecasting

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Question 31

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Random variation is one of the four different components of a time series. It is caused by irregular and unpredictable changes in a time series that are not caused by any other component. It tends to mask the existence of the other, more predictable components.

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Question 32

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Regression analysis was used to develop the following equation from 60 observations of quarterly data: $=2500-3 t-3 Q_{1}+2 Q_{2}+5 Q_{3}$ , where: $Q _ { i }$ = 1, if quarter i (i = 1, 2, 3) = 0, otherwise Forecast the next four quarters.

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Question 33

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We calculate the five-period moving average for a time series for all time periods except the:

A time series is shown in the table below: $\begin{array} { | c | c | } \hline \text { Time period } & y _ { t } \\\hline 1 & 48 \\2 & 50 \\3 & 46 \\4 & 42 \\5 & 40 \\6 & 32 \\7 & 34 \\8 & 26 \\9 & 21 \\10 & 13 \\\hline\end{array}$ a. Plot the time series to determine which of the trend models appears to fit better. b. Use the regression technique to calculate the linear trend line and the quadratic trend line. Which line fits better? Use the best model to forecast the value of y for time period 7.

A 5 period moving average will show less fluctuation than a 3 period moving average, for the same time series.

Quarterly enrolments in a business statistics class for three years are shown below. $\begin{array} { | c | c c | } \hline \text { Year } & \text { Quarter } & \text { Enrolment } \\\hline 1996 & 1 & 26 \\& 2 & 29 \\& 3 & 33 \\& 4 & 18 \\1997 & 1 & 27 \\& 2 & 25 \\& 3 & 36 \\& 4 & 21 \\& 1 & 32 \\1998 & 2 & 36 \\& 3 & 39 \\& 4 & 30 \\\hline\end{array}$ Compute the four-quarter centred moving averages.

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?

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

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. $\begin{array}{l}\text { SUMMARV OUTPUT }\\\hline\text { Regression Stotitics }\\\begin{array}{lc}\hline \text { Multiple R } & 0.9460 \\\text { R Square } & 0.8950 \\\text { Adjusted RSquare } & 0.8864 \\\text { Standard Error } & 3.7394 \\\text { Observations } & 54 \\\hline\end{array}\end{array}$ $\text { ANOVA }$ $\begin{array}{lccccc} \hline& & & & & \text { Sgnificance } \\& d f & S & M S & F & F \\\hline \text { Regression } & 4 & 5837.596003 & 1459.4 & 104.3701 & 2.41949 \mathrm{E}-23 \\\text { Residual } & 49 & 685.1632564 & 13.9829 & & \\\text { Total } & 53 & 6522.759259 & & &\\\hline\end{array}$ $\begin{array}{lcrrrrr}\hline&& \text { Standard } & & && \text { Upper } \\&\text { Coefficients } & \text { Error } & \text { t Stat } & \text {p-value } & \text { Lower } 95 \%& 95 \%\\\hline\text { Intercept } & 3.0588 & 1.3331 & 2.2944 & 0.0261 & 0.3797 & 5.7378 \\\mathrm{t} & 0.2518 & 0.0327 & 7.7052 & 0.0000 & 0.1861 & 0.3175 \\\mathrm{Q} 1 & 12.4604 & 1.3897 & 8.9664 & 0.0000 & 9.6677 & 15.2530 \\\mathrm{Q} 3 & 1.1458 & 1.4721 & 0.7784 & 0.4401 & -1.8124 & 4.1041 \\\mathrm{Q} 4 & 23.9121 & 1.4403 & 16.6025 & 0.0000 & 21.0177 & 26.8064\\\hline\end{array}$ (a) Write out the regression equations for each of the four quarters. (b) Sketch the four equations from part (a) on the same set of axes. (c) Interpret the coefficients on all the indicator variables. (c) All the indicator variables have positive coefficients. Is this surprising? Explain.

The trend equation for quarterly sales data (in millions of dollars) for 2001-2005 is $=6.8+1.2 t$ , where t = 1 for the first quarter of 2001. The seasonal index for the third quarter of 2006 is 1.25. The forecast sales for the third quarter of 2006 is:

A time series regression equation for 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?

The purpose of using the moving average is to take away the short-term seasonal and random variation, leaving behind a combined trend and cyclical movement.

The time-series component that reflects the irregular changes in a time series that are not caused by any other component, and tends to hide the existence of the other, more predictable components, is called random variation.

Which of the following is true?

Of the four components of the multiplicative time-series model, the ratio of the time series to the moving average isolates the:

Time-series forecasting with exponential smoothing uses the following formula:` $S _ { t } = w y _ { t } + ( 1 - w ) S _ { t - 1 }$ . where $S _ { t }$ is the exponentially smoothed time series at time t, $y _ { t }$ is the value of the time series at time t, and w is the smoothing constant. The forecast value at time t + 1, where w = 0.3, is given by:

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 seasonal indexes computed from the trend line for the four quarters of the year 2011 are 0.88, 0.93, 1.04, and 1.17, respectively. The seasonalised forecast for the third quarter of the year 2011 is:

One measure of the accuracy of a forecasting model is the:

Given the following time series, compute the seasonal (quarterly) indexes, using the four-quarter centred moving averages. $\begin{array} { | c | c c c c c | } \hline \text { Quarter } & 1993 & 1994 & 1995 & 1996 & 1997 \\\hline 1 & 62 & 48 & 50 & 43 & 57 \\2 & 51 & 45 & 46 & 39 & 32 \\3 & 53 & 44 & 46 & 37 & 31 \\4 & 46 & 37 & 42 & 32 & 29 \\\hline\end{array}$

Which of the following components of a time-series reflects the overall general movement of the data?