ETC3550 Applied Forecasting for Business and Economics 2019
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Semester One 2019
Examination Period
ETC3550
Applied Forecasting for Business and Economics
QUESTION 1
Write about a quarter of a page each on any FOUR of the following topics. (Clearly state if you agree or disagree with each statement. No marks will be given without any justification.)
(a) The trouble with statistical methods of forecasting is that they assume the patterns in the past will continue in the future.
(b) A time series decomposition into trend, seasonal and remainder terms is only useful when there are no cycles in the data.
(c) With STL decompositions and ETS models, we always need to transform our data before estimating the components.
(d) Some ETS models are not always suitable and should be avoided.
(e) The combination of AR and MA components guide long-run ARIMA forecasts. (f) Linear regression models are simplistic because the real world is nonlinear.
QUESTION 2
Figure 1 shows the number of employees (in thousands) in child day care services in New York City over the period the period January 1990–February 2019.
Number of employees in child day care services in New York City
30
20
(a) The following code has been used to produce Figures 1, 2and 3.
daycare %>%
autoplot (Count) +
ggtitle("Number of employees in child day care services in New York City") + xlab("Year" ) + ylab("Thousands of persons")
daycare %>%
gg_season (Count, labels= 'both ' ) +
ggtitle("Number of employees in child day care services in New York City") + ylab("Thousand of persons")
daycare %>%
gg_subseries (Count) +
ggtitle("Number of employees in child day care services in New York City") + ylab("Thousand of persons")
(b) Using the code below, describe what is plotted in Figure 4. Comment on the selection of window.
daycare %>%
model(STL (log(Count)
components() %>%
autoplot() + ggtitle("Number of
~ season(window = 21 ))) %>%
employees in child day care services in New York City")
Number of employees in child day care services in New York City ‘log(Count)‘ = trend + season_year + remainder
You are asked to provide forecasts for the next two years for the daycare series shown in Figure 1. Consider applying each of the methods and models below. Comment, in a few words each, on whether each one is appropriate for forecasting the data. No marks will be given for simply guessing whether a method or a model is appropriate without justifying your choice.
A. Seasonal naïve method.
B. Drift method plus seasonal dummies.
C. Holt-Winters additive damped trend method.
D. Holt-Winters multiplicative damped trend method.
E. ETS(A,N,M).
F. ETS(M,Ad,M).
G. ARIMA(1,1,4).
H. ARIMA(3,1,2)(1,1,0)12 .
I. ARIMA(0,1,1)(2,0,0)12 .
J. Regression model with time and Fourier terms.
QUESTION 3
The following R code and output concerns two models for the daycare series plotted in Figure 1. The estimated components of the models are plotted in Figure 5.
fit_ets <- daycare %>%
model(
trend = ETS(Count ~ trend(method = "A" )),
damped = ETS(Count ~ trend(method = "Ad" ))
)
fit_ets %>% select(trend) %>% report()
## Series: Count
## Model: ETS(M,A,M)
## Smoothing parameters:
## alpha = 0.46736
## beta = 0.00010498
## gamma = 0.35681
##
## Initial states:
## l b s1 s2 s3 s4 s5 s6 s7 s8 ## 14.013 0.061175 1.0192 1.0275 1.0127 1.0027 0.95847 0.96163 1.0028 0.9928 ## s9 s10 s11 s12
## 1.0102 1.0171 1.0017 0.99314
##
## sigma^2: 2e-04
##
## AIC AICc BIC
## 1198.8 1200.6 1264.4
fit_ets %>% select(damped) %>% report()
## Series: Count
## Model: ETS(M,Ad,M)
## Smoothing parameters:
## alpha = 0.89049
## beta = 0.015802
## gamma = 0.069298
## phi = 0.98
##
## Initial states:
## l b s1 s2 s3 s4 s5 s6 s7 s8 ## 13.545 0.045311 1.022 1.0193 1.0126 0.98462 0.92759 0.94846 1.0018 1.0164 ## s9 s10 s11 s12
## 1.0204 1.0211 1.014 1.0118
##
## sigma^2: 1e-04
##
## AIC AICc BIC
## 1147.3 1149.4 1216.8
30
20
35
30
25
20
15
0.06 0.05 0.04 0.03 0.02 0.01
1.025
1.000
0.975
0.950
0.925
0.04
0.02
0.00
−0.02
ETS(M,Ad,M) decomposition
Components
Figure 5:
(a) Comment on the diferences between the two model specifications.
(b) Comment on Figure 5and how this relates to the estimated parameters of the models.
(c) Figures 6and 7and the R-output below these relate to the residuals from the two estimated models. Comment on these in relation to the fit of the models. Give as many details as you can. What do your conclusions here imply about using these models for forecasting?
fit_ets %>%
select(trend) %>%
gg_tsresiduals()60
0.2
0.0
−0.2
−0.03
lag [1M]
Figure 6:
fit_ets %>%
select(trend) %>%
augment() %>%
features(.resid, ljung_box, lag = 24 , dof = 16 )
## .model lb_stat lb_pvalue ##
fit_ets %>%
select(damped) %>%
60
0.2
0.1
0.0
−0.1
0
−0.02
lag [1M]
Figure 7:
fit_ets %>%
select(damped) %>%
augment() %>%
features(.resid, ljung_box, lag = 24 , dof = 17 )
## # A tibble: 1 x 3
## .model lb_stat lb_pvalue
##
## 1 damped 98.1 0
(d) Considering all the analysis so far, which model would you choose for forecasting and why?
(e) Figure 8shows forecasts from the two models (in order to improve visualization only data
from 2000 onwards is included in the plots). Comment on the two sets of forecasts. Based on these would you change your decision as to which model you would choose for forecasting.
Figure 8:
(f) Write down in full your selected estimated model.
QUESTION 4
(a) Figures 9and 10show time plots, ACFs and PACFs related to the daycare series. The variables plotted were constructed as follows.
daycare %>%
mutate(
log_count = log (Count),
diff_log_count = difference(log(Count)),
sdiff_log_count = difference(log(Count), 12 ),
diff_sdiff_log_count = difference(difference(log(Count), 12 ))
3.50
3.25
3.00
2.75
0.05
0.00
−0.05
0.12
0.08
0.04
0.00
0.04
0.00
−0.04 |
1990 Jan 2000 Jan 2010 Jan 2020 Jan Date Figure 9: |
Explain what each of the ACFs and PACFs show about the stationarity, seasonality and other features of the time series.
ACF of diff_log_count PACF of diff_log_count
0.4
0.5
0.0
0.0
−0.424
lag [1M]0.6
0.3
0.0
−0.3
6 12 18
2022-06-01