ˆ
y
t
+1
=
y
t
+
t
X
i
=2
y
i

y
i

1
t

1
for one period in the future. If we extend it further to h periods:
ˆ
y
t
+
h
=
y
t
+
h
·
t
X
i
=2
y
i

y
i

1
t

1
In essence, we are taking the last observation and adding on to it, the average change in
values over the
entire
time frame.
11.4
Model Diagnostic
We want to check for autocorrelation to check if the model is capturing patterns adequately.
Furthermore, we want to do density forecasting so we need to check if the assumptions we are
building for these forecast intervals are appropriate for the models we are estimating.
Autocorrelation
of time series process is to see the correlation with itself.
⇢
k
=
E
[(
Y
t

μ
)(
Y
t

k

μ
)]
σ
2
=
Corr
(
Y
t
, Y
t

k
)
whereby k is the lag,
μ
σ
2
are the mean and variance of the time series.
55
We can then calculate the
sample autocorrelation
.
⌧
k
=
T

k
P
t
=1
(
y
t

k

¯
y
)(
y
t

¯
y
)
T
P
t
=1
(
y

¯
y
)
2
Corr(
Y
t
, Y
t

k
) =
Cov
(
Y
t
,Y
t

k
)
p
Y
t
p
Y
t

k
We make the assumption that the variance of the time series is constant.
From all this,
the
autocorrelation function (ACF)
plot tells us the autocorrelation for a range of
lags.
White Noise Process
Is a sequence of iid random variables
⇠
(0,
σ
2
). We assume that
the error terms are white noises.
If the model is well specified and captures the time series patterns well, the residuals should
behave like a white noise process.
Y
t
=
f
(
Y
1:
t
) +
✏
t
✏
t
=
f
(
Y
1:
t
)

Y
t
so if the error term is autocorrelated, then our model we set up is missing time series
pattern and therefore should adjust the model accordingly to get an uncorrelated series of
residuals.
3 diagnostic plots we should use:
Residual Plot
: The presence of patterns in time series of residuals may suggest assumption
violations and need for alternative models.
Residual ACF Plot
: This tells us the autocorrelation for di
ff
erent lag lengths. If sub
stantial autocorrelation, we have an issue.
This is for us to get accurate point forecast.
Well specified models should lead to small and insignificant sample autocorrelation, which
is consistent with a white noise process.
Residual distribution plots
:
This is useful for density forecast.
We can use normal
assumption for density forecasting but if not, we need to use bootstrapping and other
techniques. This allows us to check the appropriate assumptions for interval forecasting.
We can check these via histograms, KDE, QQ plots.
LOOK AT TUTORIAL ON HOW TO DO ALL OF THESE
Time series analysis steps
56
1) Problem definition
2) EDA
3) Modelling
4) Model Diagnostic
5) Model Evaluation
6) Forecast
11.5
Model Validation
Recall that model validation looks at whether models we are using are suitable for our
needs. We score a given model with a measure of how well it serves its purpose. On the
other hand, model selection answers which model is most useful for our needs.
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 Fall '19
 Linear Regression, Regression Analysis