COMM 2502 Dalhousie University Week 11 Time Series via Excel Minitab Practice: Statistics Answers 2021

COMM 2502 Dalhousie University Week 11 Time Series via Excel Minitab Practice: Statistics Answers 2021

COMM 2502 Dalhousie University Week 11 Time Series via Excel Minitab Practice: Statistics Answers 2021

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COMM 2502 Dalhousie University Week 11 Time Series via Excel Minitab Practice

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Time series analysis, Part 2
Data can be found in the worksheet “Week 11 – Data.xlsx”.
1. Automobile unit sales at B.J. Scott Motors, Inc., provided the following 10-year time
series.
Year
Sales
Year
Sales
1
400
6
260
2
390
7
300
3
320
8
320
4
340
9
340
5
270
10
370
a. Construct a time series plot. Comment on the appropriateness of a linear trend.
b. Using Minitab’s procedure Stat > Time Series > Trend analysis, develop a
quadratic trend equation that can be used to forecast sales and forecast sales in
year 11.
c. Set up the same analysis as a proper regression model. (Prepare proper variables
to estimate a quadratic model and make sure to request computation of the
Durbin-Watson statistic — under “Results…” and obtain residual plots.)
d. Is the model statistically significant? (Use  = 0.05)
e. Obtain a point forecast and a 95% forecast interval for sales in year 11.
f. Assess the assumptions about the residuals in the model of part c).
2. Consider the following time series data. Complete questions using Minitab.
Quarter
Year 1
Year 2
Year 3
1
4
6
7
2
2
3
6
3
3
5
6
4
5
7
8
a. Construct a time series plot. Does the data set exhibit a trend? (What shape is
it?) Is there a seasonal component?
b. Use indicator variables for Quarter 1, Quarter 2 and Quarter 3 to develop an
estimated regression equation to account for any seasonal and linear trend
effects in the data. Make sure to request sequential SS under “Options…” and to
put the trend variable into the model first.
c. Test the significance of the trend component at the 0.05 level.
d. Test the significance of the seasonal component using a partial F-test. ( = 0.05.)
e. Use Minitab to compute quarterly forecasts for the next four quarters (that is,
for year 4).
3. A random sample of house sales data contains the following columns: Period, Year, QTR,
and Sales. Use Minitab to complete following questions:
a. Construct a time series plot. Does the series exhibit a trend? (What shape?) Is
the error component likely to be additive or multiplicative?
b. Use indicator variables for Quarter 1, Quarter 2 and Quarter 3 to develop an
estimated regression equation to account for any seasonal and linear trend
effects in the data.
c. Assess the model assumptions for the model in part (b).
d. Test for positive autocorrelation at the 0.05 level of significance.
Year
1
2
3
4
5
6
7
8
9
10
Sales
400
390
320
340
270
260
300
320
340
370
Year
1
1
1
1
2
2
2
2
3
3
3
3
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
Value
4
2
3
5
6
3
5
7
7
6
6
8
Period
YEAR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
QTR
2007
2007
2007
2007
2008
2008
2008
2008
2009
2009
2009
2009
2010
2010
2010
2010
2011
2011
2011
2011
2012
2012
2012
2012
2013
2013
2013
2013
2014
2014
2014
2014
2015
2015
2015
2015
2016
2016
2016
2016
2017
2017
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
Sales(1m)
1.833
6.195
34.016
34.74545
23.72
40.00715
82.9883
168.8318
177.659903
173.561126
164.481455
191.724755
158.876197
228.717085
219.005638
263.52436
219.787788
244.275418
215.627478
245.755091
222.0488
220.249563
267.505866
316.743315
233.214037
277.179933
327.105733
335.18943
364.632771
389.721118
401.197965
541.215865
472.575739
516.80666
568.780706
725.854593
532.617761
609.335895
558.895127
781.455415
677.783175
735.093484
43
44
45
46
47
48
2017
2017
2018
2018
2018
2018
3
4
1
2
3
4
716.77901
921.353915
679.916669
756.091631
568.309423
544.6669
Comm 2502
Predictive Analytics
Week 11 – Time Series, part 2
Inferences and regression models
© 2021 H.I. Gassmann
A quick overview
• Components of a time series and
– Trend, seasonality, cyclical, and residual effects
• Forecasts
– Always outside the range of observed values
• Smoothing methods
– Moving averages
– Exponential smoothing
• Inferences
– Forecast intervals
• Regression models
© 2021 H.I. Gassmann
Smoothing methods and forecasting
• Averaging different periods
– Backward in time or centred; constant length or varying length; equal weights or unequal weights
• Simple moving averages (computing vs. forecasting)
– Average of k adjacent periods relative to reference period t
• Backward MA: t, t–1, t–2, …, t–k+1: MAt(k) =
+ −1 + −2 +⋯+ − +1

– Observations lost at start
• Centred MA: t, k/2 periods backward, k/2 periods forward. If k is even, average Yt-k/2 and Yt+k/2
– Forecast for next period +1 =
+ −1 + −2 +⋯+ − +1

• Use same forecast for all periods beyond range of data

= MA ( )
Assumes no trend, no seasonality
• Simple (single) exponential smoothing
– Implicitly combine all past data with diminishing weights
– = + 1 − −1 with 1 = 1 and a constant between 0 and 1
– Assigns weight a to Yt, a2 to Yt–1, a3 to Yt–2, …, (1 – a – a2 – …) to Y1
• Larger a gives more weight to present, smaller a gives more weight to past
– Forecast for next period, +1 =
• Use same forecast for all periods beyond range of data
© 2021 H.I. Gassmann
Forecast errors and forecast intervals
• Forecast methods may depend on which components are present (T, S, C, I)
• Forecast errors are computed ex post
– Difference between the forecast and actual value
• Assessing forecast accuracy using one-period-ahead forecasting)
– For each period t, find forecast from information available up to t – 1
• E.g., moving averages, exponential smoothing, etc. ( +1 = MAt (or ESt) )
– Mean absolute error (Average of the absolute deviation | – |)
– Mean squared error (Average of the squared deviation ( – )2 )
– Mean absolute percentage error (Average of the relative deviation | – | / )
• Forecasts beyond range of observed values
– Smoothing methods: +1 = + 2 = … = MAT (or EST)
– The interval +1 +/- 1.96* MSE is an approximate 95% forecast interval
• (This assumes normality)
© 2021 H.I. Gassmann
Example (Wells Fargo data, PE 10)
• 30-day backward moving average
• Margin of error (95% confidence): 1.96* MSE = 4.7848897
• Lower bound = 46.1667 – 4.7849, upper bound 46.1667 + 4.7849
© 2021 H.I. Gassmann
Modelling a trend
• Example: Canadian military spending
– Predictor variable: Time
15.00
Canadian Military Spending
($B)
10.00
5.00
• Can use Year or Period (1, 2, 3, …)
• Period gives easier interpretation to constant:
0.00
1975
1980
1985
1990
– Estimated spending when period = 0 (i.e., 1979)
– Make sure to request Durbin-Watson statistic (under “Results…”)
(for Period = 15 – i.e., 1994)
© 2021 H.I. Gassmann
1995
The Durbin-Watson test
• Autocorrelation
– Residuals et-1, et in periods t – 1 , t correlated
– Compare variance of et – et-1 to variance of et
• If uncorrelated, then
σ − −1 2
σ 2
should be about 2,
because Var(et – et-1) = Var(et) + Var(et-1)
• Durbin-Watson test
– H0: consecutive residuals uncorrelated
(i.e., no autocorrelation)
– Ha: there is autocorrelation (usually positive)
• Test statistic:
σ − −1 2
σ 2
• Reject H0 based on table values
© 2021 H.I. Gassmann
Tables for Durbin-Watson test
Critical Values for the Durbin-Watson test
Significance levels for a one-sided test, a = 0.05
Ho: No autocorrelation
Ha: Positive autocorrelation
Structure of the table
Rows: number of observations
Columns: Number of predictors
Decision rule
If test statistic < dL, reject Ho. There is positive AC If test statistic > dU, do not reject Ho
If between dL and dU, test is inconclusive
Example: Military spending
n = 14; p = 1; test statistic (Minitab): 0.491121
Less than 0.97, so reject Ho. Autocorrelation exists
For a two-sided test, significance level is 0.10
Rejection region: DW < dL OR DW > 4 – dL
Non-rejection region: DW > dU or < 4 – dU Region of ambiguity: between dL and dU AND between 4 – dU and 4 - dL Example: Semi-annual Walmart revenues Fitting a cubic trend model has n = 25, p = 3 DW reported as 3.06, which exceeds 4 – 1.12 = 2.88 Conclusion for Ho: Autocorrelation (positive or neg.) Reject Ho at 0.10 level 1 dL dU 6 0.61 1.40 7 0.70 1.36 8 0.76 1.33 9 0.82 1.32 0.88 1.32 10 12 0.97 1.33 15 1.08 1.36 20 1.20 1.41 25 1.29 1.45 30 1.35 1.49 35 1.40 1.52 40 1.44 1.54 45 1.48 1.57 50 1.50 1.58 1.55 1.62 60 70 1.58 1.64 80 1.61 1.66 90 1.63 1.68 100 1.65 1.69 150 1.72 1.75 200 1.76 1.78 250 1.78 1.80 300 1.80 1.82 400 1.83 1.84 500 1.85 1.86 600 1.86 1.87 800 1.88 1.89 © 2021 1.89 1.90 1000H.I. Gassmann 1500 1.91 1.92 2000 1.93 1.93 n 2 3 4 5 dL dU dL dU dL dU dL dU 0.47 0.56 0.63 0.70 0.81 0.95 1.10 1.21 1.28 1.34 1.39 1.43 1.46 1.51 1.55 1.59 1.61 1.63 1.71 1.75 1.78 1.80 1.83 1.85 1.86 1.88 1.89 1.91 1.92 1.90 1.78 1.70 1.64 1.58 1.54 1.54 1.55 1.57 1.58 1.60 1.61 1.63 1.65 1.67 1.69 1.70 1.72 1.76 1.79 1.81 1.82 1.85 1.86 1.87 1.89 1.90 1.92 1.93 0.37 0.45 0.53 0.66 0.81 1.00 1.12 1.21 1.28 1.34 1.38 1.42 1.48 1.52 1.56 1.59 1.61 1.69 1.74 1.77 1.79 1.82 1.84 1.86 1.88 1.89 1.91 1.92 2.29 2.13 2.02 1.86 1.75 1.68 1.65 1.65 1.65 1.66 1.67 1.67 1.69 1.70 1.72 1.73 1.74 1.77 1.80 1.82 1.83 1.85 1.87 1.88 1.89 1.90 1.92 1.93 0.30 0.38 0.51 0.69 0.89 1.04 1.14 1.22 1.28 1.34 1.38 1.44 1.49 1.53 1.57 1.59 1.68 1.73 1.76 1.78 1.82 1.84 1.85 1.87 1.89 1.91 1.92 2.59 2.41 2.18 1.98 1.83 1.77 1.74 1.73 1.72 1.72 1.72 1.73 1.74 1.74 1.75 1.76 1.79 1.81 1.83 1.84 1.86 1.87 1.88 1.89 1.90 1.92 1.93 0.24 0.38 0.56 0.79 0.95 1.07 1.16 1.23 1.29 1.33 1.41 1.46 1.51 1.54 1.57 1.66 1.72 1.75 1.78 1.81 1.83 1.85 1.87 1.89 1.91 1.92 2.82 2.51 2.22 1.99 1.89 1.83 1.80 1.79 1.78 1.77 1.77 1.77 1.77 1.78 1.78 1.80 1.82 1.83 1.84 1.86 1.87 1.88 1.90 1.91 1.92 1.93 What does Durbin-Watson test “prove”? Mis-specified trend Cyclical effect Durbin-Watson test cannot distinguish the two situations © 2021 H.I. Gassmann Youth unemployment, a closer look • • • • • • • No trend, no seasonality Strong cyclical effect Sample mean 14.515 Sample standard deviation 2.133 Residuals et: Observation – Mean Correlation between et and et -1: 0.6676 Use et -1 as a predictor! © 2021 H.I. Gassmann Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Data 12.2 13.7 14.0 12.6 12.8 12.7 18.2 19.2 17.3 16.2 14.8 13.2 11.5 11.0 12.4 15.8 17.1 17.1 15.8 14.7 15.3 16.2 15.1 14.0 12.6 12.8 13.6 Resid Lagged -2.3148 -0.8148 -2.3148 -0.5148 -0.8148 -1.9148 -0.5148 -1.7148 -1.9148 -1.8148 -1.7148 3.6852 -1.8148 4.6852 3.6852 2.7852 4.6852 1.6852 2.7852 0.2852 1.6852 -1.3148 0.2852 -3.0148 -1.3148 -3.5148 -3.0148 -2.1148 -3.5148 1.2852 -2.1148 2.5852 1.2852 2.5852 2.5852 1.2852 2.5852 0.1852 1.2852 0.7852 0.1852 1.6852 0.7852 0.5852 1.6852 -0.5148 0.5852 -1.9148 -0.5148 -1.7148 -1.9148 -0.9148 -1.7148 Canadian military spending, revisited • Linear trend 15.00 – Predictor variable: Time Period (1, 2, 3, …, 14) Canadian Military Spending ($B) 10.00 5.00 0.00 1975 • Quadratic trend – Predictors: Period, Period^2 (for 1994 – i.e., Period = 15, Period2 = 225) © 2021 H.I. Gassmann 1980 1985 1990 1995 Walmart: Exponential trend for annual data Walmart Annual Sales ($B) • First try: Fit a linear trend 400 300 200 100 • Second try: quadratic trend • Third try: log transform • Fourth try: Log with quadratic • Final model: Add previous period’s residuals as predictor – Justification: Residuals correlated with previous period’s residuals – Results see next slide © 2021 H.I. Gassmann 0 1970 1980 1990 2000 2010 The final model for annual Walmart data • Response: Ln(Revenue) • Predictors – Period, Period^2, Lagged residuals • Results • Large VIF: Multicollinearity – Period and Period^2 (expected) • Anderson-Darling: p-value = 0.887 • Forecasting next period – Period = 31, Period^2 = 961 – Last know residual: 2.0494 • Revenue = 399.33 – (355.62. 430.63) © 2021 H.I. Gassmann Quarterly data with indicator variables (Walmart) • First model – Predictors: Period, 3 indicators for quarters Quarter Sales ($B) Period Ind_Q3 2008, Q3 98.34 0.25 1 2008, Q4 108.63 0.5 0 2009, Q1 94.24 0.75 0 2009, Q2 100.88 1 0 … … … … Ind_Q4 0 1 0 0 … Ind_Q1 0 0 1 0 … – Note data values: Periods increment by 0.25; base case is quarter 2 • Coefficient of periods: Estimated annual increase • Intercept: All predictors are zero: 2008, Q2 – Anderson-Darling Normality test: p < 0.005 © 2021 H.I. Gassmann Quarterly data with indicator variables (cont’d) • Second model – Period, indicators for Q3, Q4, Q1, lagged residuals © 2021 H.I. Gassmann Purchase answer to see full attachment [/vc_column_text][vc_message message_box_color="success" icon_fontawesome="fas fa-check-circle"]This question was handled by a Studyhelp247 Statistics tutor and the student left a positive review. 19+ custom answers have been given to students for this question in [month] [year] alone. 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