Ka-fu Wong University of Hong Kong

1
Ka-fu WongUniversity of Hong Kong
Modeling and Forecasting Trends
2
Background

• The unobserved components approach to modeling
  and forecasting economic time series assumes that
  the typical economic time series, yt, is made up
  of the sum of three independent components
• a time trend component
• a seasonal component
• an irregular or cyclical component.
• yt time trend seasonal cyclical Tt
  St Ct
• The time trend refers to the long-run average
  behavior of the series.
• The seasonal refers to the annual predictable
  cyclical behavior of the series associated with
  weather patterns, holiday patterns, etc.
• The cyclical component refers to the remainder of
  the series after the trend and seasonal have been
  accounted for.

3
Background

• The assumption that these components are
  determined independently means that each
  component is determined and influenced by its own
  set of forces and, consequently, each component
  can be studied separately.
• The approach is called an unobserved components
  approach because we do not directly observe each
  of the three components we only get to observe
  their sum. Our job will be to model and estimate
  the various components and use these estimates as
  the basis for forecasting the components and
  their sum.

4
Background

• Whether the assumption underlying the unobserved
  components approach, that the trend, seasonal,
  and cyclical components are determined
  independently, is plausible or not is debatable
  and is, in fact, an issue of some controversy
  among economists.
• For example, many macroeconomists argue that
  economic growth (trend) and the business cycle
  (cyclical) are determined by a common set of
  forces.

5
U.S. Female Labor Force Participation Rate
6
U.S. Male Labor Force Participation Rate
7
Hong Kong labor force participation rates (male
and female)
8
Chinas per Capita Real GDP
9
Modeling the Trend

• If we look at Chinas per capita real GDP time
  series or any one of your time series, the first
  thing that stands out us is the obvious tendency
  of the series to grow (or, in some cases, to
  fall) over time.
• That is, it is immediately apparent from the time
  series plot that the average change in the series
  is positive (or, in some cases, negative). This
  tendency is the seriess trend.
• The simplest model of the time trend is the
  linear trend model
• Tt ß0 ß1t, t 1,,T

10
Modeling the Trend

• The simplest model of the time trend is the
  linear trend model
• Tt ß0 ß1t, t 1,,T
• The trend component is a straight line with
  intercept ß0 and slope ß1. And, T1 ß0 ß1, T2
  ß0 2ß1,,TT ß0 Tß1.
• Note that ß1 dTt/dt and ß1 Tt Tt-1. So,
• ß1 gt 0 if y has a positive trend and
• ß1lt 0 if y has a negative trend.
• The intercept, as is often the case in
  econometric models, does not have a meaningful
  interpretation and its sign can be positive or
  negative, regardless of the trends sign.

11
Graphical view of linear trend
An downward trend
An upward trend
12
Polynomial trend model

• In some cases, a linear trend is inadequate to
  capture the trend of a time series. A natural
  generalization of the linear trend model is the
  polynomial trend model
• Tt ß0 ß1t ß2t2 ßptp where p is a
  positive integer.
• Note that the linear trend model is a special
  case of the polynomial trend model (p1).
• For economic time series we almost never require
  p gt 2. That is, if the linear trend model is not
  adequate, the quadratic trend model will usually
  work
• Tt ß0 ß1t ß2t2
• In the quadratic model, dTt/dt ß12tß2

13
Graphical view of Quadratic Trends
14
Graphical view of Quadratic Trends
15
The Log Linear Trend Model

• Another alternative to the linear trend model is
  the log linear trend model, which is also called
  the exponential trend model
• Tt ß0exp(ß1t)
• or, taking natural logs on both sides,
• log(Tt) log(ß0) ß1t
• so that the log of the trend component is
  linear.
• Note that for the log linear trend model
• ß1 log(Tt) log(Tt-1) change in T

16
Graphical view of exponential trends
17
Graphical view of exponential trends
18
Which trend model to use?

• Knowing the differences among these models can
  help us decide whether the linear, quadratic or
  log linear trend model is more appropriate for
  our data.
• In the linear trend model the change in T is
  constant over time.
• In the quadratic trend model the change in T has
  a linear trend.
• In the log linear trend model the growth rate
  that is constant over time.
• However, in practice, it is not always obvious by
  simply looking at the time series plot which form
  the trend model should take linear, log linear,
  quadratic? Other?
• Practice and experience are the most helpful.

19
All Deterministic Trend Models

• Note that in all of these models, the trend is
  deterministic, i.e., perfectly forecastable. For
  instance, in the linear trend model, the forecast
  of TTh made at time T is
• ß0 (Th)ß1 TTh
• (Later in the course we will talk about
  stochastic trend models, in which the trend of
  the series is not perfectly forecastable.)
• However, even if we correctly specify the shape
  of the trend (linear, quadratic, exponential, ),
  the parameters of the trend model are unknown.
  So, in practice, we will have to estimate these
  parameters, which will introduce errors (called
  sampling or estimation error) into our trend
  forecasts.

20
Estimating the Trend Model

• Our assumption at this point is that our time
  series, yt, can be modeled as
• yt Tt(?) et
• where
• Tt is one of the trend models we discussed
  earlier,
• ? is the set of parameters ? (ß0, ß1) in a
  linear trend model.
• et denotes the other factors (i.e., the seasonal
  and cyclical components) that determine yt.
• Since ? is unknown, it is natural to estimate the
  trend model via the least squares approach

Quadratic loss
The choice of ? that will minimize the objective
function.
21
Estimating the Trend Model via the Least Squares
approach

• For linear trend model

can use OLS

• For quadratic trend model

can use OLS

• For exponential trend model

Nonlinear, has to be estimated numerically.
or
can use OLS
22
Property of the Ordinary Least Squares Estimators

• Under the assumptions of the unobserved
  components model, the OLS estimator of the linear
  and quadratic trend models is
• unbiased,
• consistent, and
• asymptotically efficient.
• Standard regression procedures can be applied to
  test hypotheses about the ?s and construct
  interval estimates. This is true even though the
  regression errors will generally be serially
  correlated and heteroskedastic.

23
Forecasting the Trend

• Once we have specified a trend model our forecast
  of the h-step ahead trend component of y will
  simply be
• Tth(?)
• When ? is unknown, we can estimate it as
  discussed earlier. And, substitute the estimate
  into the function above.

24
Forecasting the Trend
We would like to forecast yTh based on all
information available at time T.

• Assume that the trend is linear.

If we know the true parameters, the part
?0?1TIMETh can be forecasted perfectly.
Can we forecast eTh? Sometimes YES. Sometimes
NO.
NO when et is known to be an independent
zero-mean random noise.
If et is an i.i.d. sequence with zero mean then
E(eTh ? information available at time T)
E(eTh) E(et) 0.
independent
identical
zero mean
25
Forecasting the Trend
Assume et is known to be an independent zero-mean
random noise.
Forecast when parameters are known
Emphasize that forecast is made at time T,
utilizing all information that is available at
time T (usually all past information).
Forecast error
Fundamental uncertainty! Unavoidable !!
Forecast when parameters are unknown
Substitute in the estimate from the OLS
regression.
Forecast error
Due to parameter uncertainty. (increases with h)
Note TIMETh Th
26
Density forecast

• Suppose we have no parameter uncertainty, we have

and
The forecast error

• Then the distribution of the forecast error will
  simply be the distribution of ?Th. That is, for
  any real number c,

E(eTh,T) 0,Var(eTh,T) ?2, where ?2
var(?t).
27
Density forecast

• Further assume the ?s are i.i.d. N(0,?2), while
  continuing to ignore parameter uncertainty. Then
  the density forecast will be that

• Note that this density forecast depends on the
  unknown parameter ?2. To make the density
  forecast operational, we can replace ?2 with an
  unbiased and consistent estimator,

28
Density forecast

• Now consider the case with parameter uncertainty.

• Under usual assumptions, the forecast error due
  to parameter uncertainty is asymptotically normal.

• Thus, eTh,T will be asymptotically normal.

• The unknown variance may be estimated as

Because we assume a linear trend.
29
(No Transcript)
30
Density forecast
Then we act as though yTh is distributed as
or, equivalently,
So, for example,


where Z is an N(0,1) random variable.
31
Density forecast

• Further, we can construct interval forecasts of
  yTh according to

is a (1-?)100 forecast interval for yTh,
where Z1-(?/2) is the (1-(?/2))100 percentile
of the N(0,1) distribution.

• For example, if ? .05 then we obtain a
  95-percent forecast interval for yTh,

since 1.96 is the 97.5 percentile of the N(0,1).

• Recall the interpretation of this kind of
  interval 95 of the time, this procedure will
  produce an interval that will turn out to include
  the actual value of yTh.

32
Selecting Forecasting ModelsR-square as a
criteria

• Consider the mean squared error (MSE)

where T is the sample size and

• Note that models with smallest MSE is also the
  model with smallest sum of squared residuals,
  because scaling the sum of squared residuals with
  a constant (1/T) will not change the ranking.

33
Selecting Forecasting ModelsR-square as a
criteria

• Consider the R-square (R2)

Depends only on data, not on model.

• Thus, models with the largest R-square is also
  the model with the smallest MSE, and also the
  model with smallest sum of squared residuals,
  because scaling the sum of squared residuals by a
  model-independent quantity will not change the
  ranking.

34
Selecting Forecasting ModelsR-square as a
criteria

• The R-square (R2) may be a good measure of
  in-sample fit but a bad measure for out-of-sample
  fit.

• Add an additional regressor in the model, we will
  always obtain a R-square (R2) no less than the
  one with less regressors. That is, a polynomial
  trend model with a larger p will almost always
  result in a smaller MSE and hence a larger
  R-square.

• In fact, give me a time series and specify an R2,
  subject to data availability, I can almost always
  produce a trend model that will attain the
  specific R2.
• This effect is called in-sample overfitting or
  data mining.

35
Selecting Forecasting ModelsR-square as a
criteria

• In short, the MSE is a biased estimator of
  out-of-sample h-step-ahead prediction error
  variance.
• because the forecast error consists of two
  parts
• Fundamental uncertainty (unavoidable even if we
  know the parameters)
• Parameter uncertainty (increases with the number
  of parameters in the model)

• To reduce the bias associated with MSE and
  R-square, we need to penalize for the number of
  parameters included in the model (or the degree
  of freedom).

36
Selecting Forecasting ModelsAdjusted R-square as
a criteria

• Adjusted R-square

Number of parameters or degree of freedom

• Maximizing adjusted R-square is like minimizing
  s2.

S2 increases with number of parameters.
37
Selecting Forecasting ModelsCriteria that
penalize number of model parameters

• Akaike information criterion (AIC)

• Schwarz information criterion (SIC)

38
The variation of criteria with k/T
39
Use the consistent model selection criteria

• A model selection criterion is consistent if the
  following conditions are met
• When the true model i.e., the data-generating
  process (DGP) is among the models considered,
  the probability of selecting the true DGP
  approaches 1 as the sample size gets large.
• When the true model is not among those
  considered, so that it is impossible to select
  the true DGP, the probability of selecting the
  best approximation to the true DGP approaches 1
  as the sample size gets large.
• SIC is consistent but AIC is not.

40
Use the asymptotically efficient model selection
criteria

• A asymptotically efficient model selection
  criterion chooses a sequence of models, as the
  sample size get large, whose 1-step-ahead
  forecast error variances approach the one that
  would be obtained using the true model with known
  parameters at a rate at least as fast as that of
  any other model selection criteria.
• AIC is asymptotically efficient but SIC is not.

41
AIC or SIC

• Usually AIC and SIC suggest the same model.
• When AIC and SIC suggest different models, we
  usually choose the model selected by SIC because
  the SIC often suggests a more parsimonious model
  (i.e., smaller number of parameters).

42
AIC and SIC reported across software packages

• ln(AIC) ln(MSE) 2k/T
• ln(SIC) ln(MSE) kln(T)/T

43
Out-of-sample fitting

• The AIC and SIC are in-sample fit criteria,
  although they account for the costs of
  overfitting through the inclusion of penalty
  term.
• What we are really interested in is the question
  
• Having fit the model over the sample period, how
  well does it forecast outside of that sample?
• The in-sample fit criteria that we discussed do
  not directly answer this question.

44
Out-of-sample fitting

• Suppose we have a data sample y1,,yT.
• Break it up into two parts (where n ltlt T)
• y1,yT-n (first T-n observations)
• yT-n1,,yT (last n observations)

1
T-n
T
T-n1
Use to estimate the model
Save n observations for checking the
out-of-sample fit
45
Out-of-sample fitting

• Break it up into two parts (where n ltlt T)
• y1,yT-n (first T-n observations)
• yT-n1,,yT (last n observations)
• Fit the shortened sample, y1,,yT-n to various
  trend models that may seem like plausible choices
  based on time series plots, in-sample fit
  criteria, linear, quadratic, the one selected
  by AIC/SIC, log linear,
• For each estimated trend model, forecast
  yT-n1,,yT and compute the forecast errors
  e1,,en
• Compare the errors across the various models
• time series plots (of the forecasts and actual
  values of yT-n1,,yT of the forecast errors)
• tables of the forecasts, actuals, and errors
• mean squared prediction errors (MSPE)

46
Out-of-sample fitting

• The advantage of this approach is that we are
  actually comparing the trend models in terms of
  their out-of-sample forecasting performance.
• A disadvantage is that the comparison is based on
  models fit over T-n observations rather than the
  T observations we have available. (Note that if
  you do use this approach and, for example, settle
  on the quadratic model, then when you proceed to
  construct your forecasts for T1, you should
  use the quadratic model fit to the full T
  observations in your sample.)
• Will the fact that, for example, the quadratic
  trend model outperformed other models in
  forecasting out of sample based on the short
  sample mean that it will perform best in
  forecasting beyond the full sample? No.

47
End