Lecture 1: Introduction to Forecasting
UCSD, January 9 2017
Allan Timmermann1
1UC San Diego
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1 Course objectives
2 Challenges facing forecasters
3 Forecast Objectives: the Loss Function
4 Common Assumptions on Loss
5 Specific Types of Loss Functions
6 Multivariate loss
7 Does the loss function matter?
8 Informal Evaluation Methods
9 Out-of-Sample Forecast Evaluation
10 Some easy and hard to predict variables
11 Weak predictability but large economic gains
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Course objectives: Develop
Skills in analyzing, modeling and working with time series data from
finance and economics
Ability to construct forecasting models and generate forecasts
formulating a class of models using information intelligently
model selection
estimation making best use of historical data
Develop creativity in posing forecasting questions, collecting and
using often incomplete data
which data help me build a better forecasting model?
Ability to critically evaluate and compare forecasts
reasonable (simple) benchmarks
skill or luck? Overfitting (data mining)
Compete or combine?
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Ranking forecasters: Mexican inflation
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Forecast situations
Forecasts are used to guide current decisions that affect the future
welfare of a decision maker (forecast user)
Predicting my grade updating information on the likely grade as the
course progresses
Choosing between a fixed-rate mortgage (interest rate fixed for 20
years) versus a floating-rate (variable) mortgage
Depends on interest rate and inflation forecast
Political or sports outcomes prediction markets
Investing in the stock market. How volatile will the stock market be?
Predicting Chinese property prices. Supply and demand considerations,
economic growth
Structural versus reduced-form approaches
Depends on the forecast horizon: 1 month vs 10 years
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Forecasting and decisions
Credit card company deciding which transactions are potentially
fraudulent and should be denied (in real time)
requires fitting a model to past credit card transactions
binary data (zero-one)
Central Bank predicting the state of the economy timing issues
Predicting which fund manager (if any) or asset class will outperform
Forecasting the outcome of the world cup:
http://www.goldmansachs.com/our-thinking/outlook/world-cup-
sections/world-cup-book-2014-statistical-model.html
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Forecasting the outcome of the world cup
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Key issues
Decision makers actions depend on predicted future outcomes
Trade off relative costs of over- or underpredicting outcomes
Actions and forecasts are inextricably linked
good forecasts are expected to lead to good decisions
bad forecasts are expected to lead to poor decisions
Forecast is an intermediate input in a decision process, rather than an
end product of separate interest
Loss function weighs the cost of possible forecast errors like a utility
function uses preferences to weigh different outcomes
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Loss functions
Forecasts play an important role in almost all decision problems where
a decision makers utility or wealth is affected by his current and
future actions and depend on unknown future events
Central Banks
Forecast inflation, unemployment, GDP growth
Action: interest rate; monetary policy
Trade off cost of over- vs. under-predictions
Firms
Forecast sales
Action: production level, new product launch
Trade off inventory vs. stock-out/goodwill costs
Money managers
Forecast returns (mean, variance, density)
Action: portfolio weights/trading strategy
Trade off Risk vs. return
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Ways to generate forecasts
Rule of thumb. Simple decision rule that is not optimal, but may be
robust
Judgmental/subjective forecast, e.g., expert opinion
Combine with other information/forecasts
Quantitative models
an estimated forecasting model provides a characterization of what
we expect in the present, conditional upon the past, from which we
infer what to expect in the future, conditional upon the present and the
past. Quite simply, we use the estimated forecasting model to
extrapolate the observed historical data. (Frank Diebold, Elements of
Forecasting).
Combine different types of forecasts
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Forecasts: key considerations
Forecasting models are simplified approximations to a complex reality
How do we make the right shortcuts?
Which methods seem to work in general or in specific situations?
Economic theory may suggest relevant predictor variables, but is silent
about functional form, dynamics of forecasting model
combine art (judgment) and science
how much can we learn from the past?
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Forecast object what are we trying to forecast?
Event outcome: predict if a certain event will happen
Will a bank or hedge fund close?
Will oil prices fall below $40/barrel in 2017?
Will Europe experience deflation in 2017?
Event timing: it is known that an event will happen, but unknown
when it will occur
When will US stocks enter a bear market (Dow drops by 10%)?
Time-series: forecasting future values of a continuous variable by
means of current and past data
Predicting the level of the Dow Jones Index on March 15, 2017
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Forecast statement
Point forecast
Single number summarizing best guess. No information on how
certain or precise the point forecast is. Random shocks affect all
time-series so a non-zero forecast error is to be expected even from a
very good forecast
Ex: US GDP growth for 2017 is expected to be 2.5%
Interval forecast
Lower and upper bound on outcome. Gives a range of values inside
which we expect the outcome will fall with some probability (e.g., 50%
or 95%). Confidence interval for the predicted variable. Length of
interval conveys information about forecast uncertainty.
Ex: 90% chance US GDP growth will fall between 1% and 4%
Density or probability forecast
Entire probability distribution of the future outcome
Ex: US GDP growth for 2017 is Normally distributed N(2.5,1)
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Forecast horizon
The best forecasting model is likely to depend on whether we are
forecasting 1 minute, 1 day, 1 month or 1 year ahead
We refer to an hstep-ahead forecast, where h (short for horizon)
is the number of time periods ahead that we predict
Often you hear the argument that fundamentals matter in the long
run, psychological factors are more important in the short run
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Information set
Do we simply use past values of a series itself or do we include a
larger information set?
Suppose we wish to forecast some outcome y for period T + 1 and
have historical data on this variable from t = 1, ..,T . The univariate
information set consists of the series itself up to time T :
IunivariateT = {y1, , yT }
If data on other series, zt (typically an N 1 vector), are available,
we have a multivariate information set
ImultivariateT = {y1, , yT , z1, , zT }
It is often important to establish whether a forecast can benefit from
using such additional information
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Loss function: notations
Outcome: Y
Forecast: f
Forecast error: e = Y f
Observed data: Z
Loss function: L(f ,Y ) R
maps inputs f ,Y to the real number line R
yields a complete ordering of forecasts
describes in relative terms how costly it is to make forecast errors
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Loss Function Considerations
Choice of loss function that appropriately measures trade-offs is
important for every facet of the forecasting exercise and affects
which forecasting models are preferred
how parameters are estimated
how forecasts are evaluated and compared
Loss function reflects the economics of the decision problem
Financial analystsforecasts; Hong and Kubik (2003), Lim (2001)
Analysts tend to bias their earnings forecasts (walk-down effect)
Sometimes a forecast is best viewed as a signal in a strategic game
that explicitly accounts for the forecast providers incentives
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Constructing a loss function
For profit maximizing investors the natural choice of loss is the
function relating payoffs (through trading rule) to the forecast and
realized returns
Link between loss and utility functions: both are used to minimize risk
arising from economic decisions
Loss is sometimes viewed as the negative of utility
U(f ,Y ) L(Y , f )
Majority of forecasting papers use simple off the shelfstatistical loss
functions such as Mean Squared Error (MSE)
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Common Assumptions on Loss
Granger (1999) proposes three requiredproperties for error loss
functions, L(f , y) = L(y f ) = L(e):
A1. L(0) = 0 (minimal loss of zero for perfect forecast);
A2. L(e) 0 for all e;
A3. L(e) is monotonically non-decreasing in |e| :
L(e1) L(e2) if e1 > e2 > 0
L(e1) L(e2) if e1 < e2 < 0A1: normalizationA2: imperfect forecasts are more costly than perfect onesA3: regularity condition – bigger forecast mistakes are (weakly)costlier than smaller mistakes (of same sign)Timmermann (UCSD) Forecasting Winter, 2017 18 / 64Additional Assumptions on LossSymmetry:L(y f , y) = L(y + f , y)Granger and Newbold (1986, p. 125): .. an assumption of symmetryabout the conditional mean … is likely to be an easy one to accept …an assumption of symmetry for the cost function is much lessacceptable.Homogeneity: for some positive function h(a) :L(ae) = h(a)L(e)scaling doesnt matterDifferentiability of loss with respect to the forecast (regularitycondition)Timmermann (UCSD) Forecasting Winter, 2017 19 / 64Squared Error (MSE) LossL(e) = ae2, a > 0
Satisfies the three Granger properties
Homogenous, symmetric, differentiable everywhere
Convex: penalizes large forecast errors at an increasing rate
Optimal forecast:
f = arg
f
min
(y f )2pY dy
First order condition
f =
ypY dy = E (y)
The optimal forecast under MSE loss is the conditional mean
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Piece-wise Linear (lin-lin) Loss
L(e) = (1 )e1e>0 e1e0, 0 < < 11e>0 = 1 if e > 0, otherwise 1e>0 = 0. Indicator variable
Weight on positive forecast errors: (1 )
Weight on negative forecast errors:
Lin-lin loss satisfies the three Granger properties and is homogenous
and differentiable everywhere with regard to f , except at zero
Lin-lin loss does not penalize large errors as much as MSE loss
Mean absolute error (MAE) loss arises if = 1/2:
L(e) = |e|
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MSE vs. piece-wise Linear (lin-lin) Loss
-3 -2 -1 0 1 2 3
0
5
10
L(
e)
e
= 0.25
-3 -2 -1 0 1 2 3
0
5
10
L(
e)
e
= 0.5, MAE loss
-3 -2 -1 0 1 2 3
0
5
10
L(
e)
e
= 0.75
MSE
linlin
MSE
linlin
MSE
linlin
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Optimal forecast under lin-lin Loss
Expected loss under lin-lin loss:
EY [L(Y f )] = (1 )E [Y |Y > f ] E [Y |Y f ]
First order condition:
f = P1Y (1 )
PY : CDF of Y
The optimal forecast is the (1 ) quantile of Y
= 1/2 : optimal forecast is the median of Y
As increases towards one, the optimal forecast moves further to the
left of the tail of the predicted outcome distribution
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Optimal forecast of N(0,1) variable under lin-lin loss
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
f*
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Linex Loss
L(e) = exp(a2e) a2e 1, a2 6= 0
Differentiable everywhere
Asymmetric: a2 controls both the degree and direction of asymmetry
a2 > 0 : loss is approximately linear for e < 0 and approximatelyexponential for e > 0
Large underpredictions are very costly (f < y , so e = y f > 0)
Converse is true when a2 < 0Timmermann (UCSD) Forecasting Winter, 2017 25 / 64MSE versus Linex Loss-3 -2 -1 0 1 2 305101520L(e)eright-skewed linex loss with a2=1-3 -2 -1 0 1 2 305101520L(e)eleft-skewed linex loss with a2=-1MSELinexMSELinexTimmermann (UCSD) Forecasting Winter, 2017 26 / 64Linex LossSuppose Y N(Y , 2Y ). ThenE [L(e)] = exp(a2(Y f ) +a2222Y ) a2(Y f )Optimal forecast:f = Y +a222YUnder linex loss, the optimal forecast depends on both the mean andvariance of Y (Y and 2Y ) as well as on the curvature parameter ofthe loss function, a2Timmermann (UCSD) Forecasting Winter, 2017 27 / 64Optimal bias under Linex Loss for N(0,1) variable-3 -2 -1 0 1 2 300.20.4eMSE loss-3 -2 -1 0 1 2 300.20.4elinex loss with a2=1-3 -2 -1 0 1 2 300.20.4elinex loss with a2=-1Timmermann (UCSD) Forecasting Winter, 2017 28 / 64Multivariate Loss FunctionsMultivariate MSE loss with n errors e = (e1, …, en) :MSE (A) = e AeA is a nonnegative and positive definite n n matrixThis satisfies the basic assumptions for a loss functionWhen A = In, covariances can be ignored and the loss functionsimplifies to MSE (In) = E [e e] = ni=1 e2i , i.e., the sum of theindividual mean squared errorsTimmermann (UCSD) Forecasting Winter, 2017 29 / 64Does the loss function matter?Cenesizoglu and Timmermann (2012) compare statistical andeconomic measures of forecasting performance across a large set ofstock return prediction models with time-varying mean and volatilityEconomic performance is measured through the certainty equivalentreturn (CER), i.e., the risk-adjusted returnStatistical performance is measured through mean squared error (MSE)Performance is measured relative to that of a constant expected return(prevailing mean) benchmarkCommon for forecast models to produce worse mean squared error(MSE) but better return performance than the benchmarkRelation between statistical and economic measures of forecastingperformance can be weakTimmermann (UCSD) Forecasting Winter, 2017 30 / 64Does loss function matter? Cenesizoglu and TimmermannTimmermann (UCSD) Forecasting Winter, 2017 31 / 64Percentage of models with worse statistical but bettereconomic performance than prevailing mean (CT, 2012)CER is certainty equivalent returnSharpe is the Sharpe ratioRAR is risk-adjusted returnRMSE is root mean squared (forecast) errorTimmermann (UCSD) Forecasting Winter, 2017 32 / 64Example: Directional Trading systemConsider the decisions of a risk-neutral market timerwhose utility islinear in the return on the market portfolio (y)U((f ), y) = yInvestors decision rule, (f ) : go longone unit in the risky asset if apositive return is predicted (f > 0), otherwise go short one unit:
(f ) =
{
1 if f 0
1 if f < 0Let sign(y) = 1, if y > 0, otherwise sign(y) = 0. Payoff:
U(y , (f )) = (2sign(f ) 1)y
Sign and magnitude of y and sign of f matter to traders utility
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Example: Directional Trading system (cont.)
Which forecast approach is best under the directional trading rule?
Since the trader ignores information about the magnitude of the
forecast, an approach that focuses on predicting only the sign of the
excess return could make sense
Leitch and Tanner (1991) studied forecasts of T-bill futures:
Professional forecasters reported predictions with higher mean squared
error (MSE) than those from simple time-series models
Puzzling since the time-series models incorporate far less information
than the professional forecasts
When measured by their ability to generate profits or correctly forecast
the direction of future interest rate movements the professional
forecasters did better than the time-series models
Professional forecastersobjectives are poorly approximated by MSE
loss closer to directional or signloss
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Common estimates of forecasting performance
Define the forecast error et+h|t = yt+h ft+h|t . Then
MSE = T1
T
t=1
e2t+h|t
RMSE =
T1 T
t=1
e2
t+h|t
MAE = T1
T
t=1
|et+h|t |
Directional accuracy (DA): let Ixt+1>0 = 1 if xt+1 > 0, otherwise
Ixt+1>0 = 0. Then an estimate of DA is
DA = T1
T
t=1
Iyt+hft+h|t>0
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Forecast evaluation
ft+h|t : forecast of yt+h given information available at time t
Given a sequence of forecasts, ft+h|t , and outcomes, yt+h,
t = 1, ,T , it is natural to ask if the forecast was optimalor
obviously deficient
Questions posed by forecast evaluation are related to the
measurement of predictive accuracy
Absolute performance measures the accuracy of an individual
forecast relative to the outcome, using either an economic
(loss-based) or a statistical metric
Relative performance compares the performance of one or several
forecasts against some benchmark
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Forecast evaluation (cont.)
Forecast evaluation amounts to understanding if the loss from a given
forecast is small enough
Informal methods graphical plots, decompositions
Formal methods distribution of test statistic for sample averages of
loss estimates can depend on how the forecasts were constructed, e.g.
which estimation method was used
The method (not only the model) used to construct the forecast
matters expanding vs. rolling estimation window
Formal evaluation of an individual forecast requires testing whether
the forecast is optimal with respect to some loss function and a
specific information set
Rejection of forecast optimality suggests that the forecast can be
improved
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Effi cient Forecast: Definition
A forecast is effi cient (optimal) if no other forecast using the available
data, xt It , can be used to generate a smaller expected loss
Under MSE loss:
f t+h|t = arg
f (xt )
minE
[
(yt+h f (xt ))2
]
If we can use information in It to produce a more accurate forecast,
then the original forecast would be suboptimal
Effi ciency is conditional on the information set
weak form forecast effi ciency tests include only past forecasts and
past outcomes It = {yt , yt1, , ft |t1, et |t1, }
strong form effi ciency tests extend this to include all other variables
xt It
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Optimality under MSE loss
First order condition for an optimal forecast under MSE loss:
E [
(yt+h ft+h|t )2
ft+h|t
] = 2E
[
yt+h ft+h|t
]
= 2E
[
et+h|t
]
= 0
Similarly, conditional on information at time t, It :
E [et+h|t |It ] = 0
The expected value of the forecast error must equal zero given
current information, It
Test E [et+h|txt ] = 0 for all variables xt It known at time t
If the forecast is optimal, no variable known at time t can predict its
future forecast error et+h|t . Otherwise the forecast wouldnt be
optimal
If I can predict that my forecast will be too low, I should increase my
forecast
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Optimality properties under Squared Error Loss
1 Optimal forecasts are unbiased: the forecast error et+h|t has zero
mean, both conditionally and unconditionally:
E [et+h|t ] = E [et+h|t |It ] = 0
2 h-period forecast errors (et+h|t) are uncorrelated with information
available at the time the forecast was computed (It). In particular,
single-period forecast errors, et+1|t , are serially uncorrelated:
E [et+1|tet |t1] = 0
3 The variance of the forecast error (et+h|t) increases (weakly) in the
forecast horizon, h :
Var(et+h+1|t ) Var(et+h|t ) for all h 1
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Optimality properties under Squared Error Loss (cont.)
Forecasts should be unbiased. Why? If they were biased, we could
improve the forecast simply by correcting for the bias
Suppose ft+1|t is biased:
yt+1 = 1+ ft+1|t + t+1, t+1 WN(0,
2)
The bias-corrected forecast:
f t+1|t = 1+ ft+1|t
is more accurate than ft+1|t
Forecast errors should be unpredictable:
Suppose yt+1 ft+1|t = et+1 = 0.5et + t+1 so the one-step forecast
error is serially correlated
Adding back 0.5et to the original forecast yields a more accurate
forecast: f t+1|t = ft+1|t + 0.5et is better than f
t+1|t
Variance of forecast error increases in the forecast horizon
We learn more information as we get closer to the forecast target
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Informal evaluation methods (Greenbook forecasts)
Time-series graph of forecasts and outcomes {ft+h|t , yt+h}Tt=1
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
-10
-5
0
5
10
GDP growth
time
an
nu
al
iz
ed
c
ha
ng
e
Actual
Forecast
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
0
2
4
6
8
10
12
14
inflation rate
time, t
an
nu
al
iz
ed
c
ha
ng
e
Actual
Forecast
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Informal evaluation methods (Greenbook forecasts)
Scatterplots of {ft+h|t , yt+h}Tt=1
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-5
0
5
10
GDP growth
forecast
ac
tu
al
0 5 10 15
0
5
10
15
inflation rate
forecast
ac
tu
al
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Informal evaluation methods (Greenbook Forecasts)
Plots of ft+h|t yt against yt+h yt : directional accuracy
-15 -10 -5 0 5 10 15
-10
-5
0
5
10
forecast
ac
tu
al
GDP growth
-10
-5
0
5
10
-15 -10 -5 0 5 10 15
-4 -3 -2 -1 0 1 2 3 4
-6
-4
-2
0
2
4
6
forecast
ac
tu
al
inflation rate
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
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Informal evaluation methods (Greenbook forecasts)
Plot of forecast errors et+h = yt+h ft+h|t
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
-5
0
5
10
fo
re
ca
st
e
rr
or
GDP growth
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
-4
-2
0
2
4
6
fo
re
ca
st
e
rr
or
time, t
Inflation rate
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Informal evaluation methods
Theil (1961) suggested the following decomposition:
E [y f ]2 = E [(y Ey) (f Ef ) + (Ey Ef )]2
= (Ey Ef )2 + (y f )2 + 2yf (1 )
MSE depends on
squared bias (Ey Ef )2
squared differences in standard deviations (y f )2
correlation between the forecast and outcome
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Pseudo out-of-sample Forecasts
Simulated (pseudo) out-of-sample (OoS) forecasts seek to mimic
the real timeupdating underlying most forecasts
What would a forecaster have done (historically) at a given point in
time?
Method splits data into an initial estimation sample (in-sample
period) and a subsequent evaluation sample (OoS period)
Forecasts are based on parameter estimates that use data only up to
the date when the forecast is computed
As the sample expands, the model parameters get updated, resulting
in a sequence of forecasts
Why do out-of-sample forecasting?
control for data mining harder to game
feasible in real time (less look-ahead bias)
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Pseudo out-of-sample forecasts (cont.)
Out-of-sample (OoS) forecasts impose the constraint that the
parameter estimates of the forecasting model only use information
available at the time the forecast was computed
Only information known at time t can be used to estimate and select
the forecasting model and generate forecasts ft+h|t
Many variants of OoS forecast estimation methods exist. These can
be illustrated for the linear regression model
yt+1 =
xt + t+1
ft+1|t =
txt
t =
(
t
s=1
(s, t)xs1x
s1
)1 (
t
s=1
(s, t)xs1y
s
)
Different methods use different weighting functions (s, t)
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Expanding window
Expanding or recursive estimation windows put equal weight on all
observations s = 1, , t to estimate the parameters of the model:
(s, t) =
{
1 1 s t
0 otherwise
As time progresses, the estimation sample grows larger, It It+1
If the parameters of the model do not change (stationarity), the
expanding window approach makes effi cient use of the data and leads
to consistent parameter estimates
If model parameters are subject to change, the approach leads to
biased forecasts
The approach works well empirically due to its use of all available
data which reduces the effect of estimation error on the forecasts
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Expanding window
1 t t+1 t+2 T-1
time
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Rolling window
Rolling window uses an equal-weighted kernel of the most recent
observations to estimate the parameters of the forecasting model
(s, t) =
{
1 t + 1 s t
0 otherwise
Only one designparameter: (length of window)
Practical way to account for slowly-moving changes to the data
generating process
Does this address breaks?
window too long immediately after breaks
window too short further away
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Rolling window
t-w+1 t-w+2 t t+1 t+2 T-1
time
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Fixed window
Fixed window uses only the first 0 observations to once and for all
estimate the parameters of the forecasting model
(s, t) =
{
1 1 s 0
0 otherwise
This method is typically employed when the costs of estimation are
very high, so re-estimating the model with new data is prohibitively
expensive or impractical in real time
The method also makes analytical results easier
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Fixed window
1 w t t+1 t+2 T-1
time
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Exponentially declining weights
In the presence of model instability, it is common to discount past
observations using weights that get smaller, the older the data
Exponentially declining weights take the following form:
(s, t) =
{
ts 1 s t
0 otherwise
0 < < 1.This method is sometimes called discounted least squares as thediscount factor, , puts less weight on past observationsTimmermann (UCSD) Forecasting Winter, 2017 55 / 64ComparisonsExpanding estimation window: number of observations availablefor estimating model parameters increases with the sample sizeEffect of estimation error gets reducedFixed/rolling/discounted window: parameter estimation errorcontinues to affect the forecasts even as the sample grows largemodel parameters are inconsistentForecasts vary more under the short (fixed and rolling) estimationwindows than under the expanding windowTimmermann (UCSD) Forecasting Winter, 2017 56 / 64US stock indexTimmermann (UCSD) Forecasting Winter, 2017 57 / 64Monthly US stock returnsTimmermann (UCSD) Forecasting Winter, 2017 58 / 64Monthly inflationTimmermann (UCSD) Forecasting Winter, 2017 59 / 64US T-bill rateTimmermann (UCSD) Forecasting Winter, 2017 60 / 64US Stock market volatilityTimmermann (UCSD) Forecasting Winter, 2017 61 / 64Example: Portfolio Choice under Mean-Variance UtilityT-bills with known payoff rf vs stocks with uncertain return rst+1 andexcess return rt+1 = r st+1 rfWt = $1 : Initial wealtht : portion of portfolio held in stocks at time t(1t ) : portion of portfolio held in TbillsWt+1 : future wealthWt+1 = (1t )rf +t (rt+1 + rf ) = rf +t rt+1Investor chooses t to maximize mean-variance utility:Et [U(Wt+1)] = Et [Wt+1]A2Vart (Wt+1)Et [Wt+1] and Vart (Wt+1) : conditional mean and variance of Wt+1Timmermann (UCSD) Forecasting Winter, 2017 62 / 64Portfolio Choice under Mean-Variance Utility (cont.)Suppose stock returns follow the processrt+1 = + xt + t+1xt (0, 2x ), t+1 (0, 2 ), cov(xt , t+1) = 0xt : predictable component given information at tt+1 : unpredictable innovation (shock)Uninformed investors (no information on xt) stock holding:t = argtmax{t+ rf A22t (2x + 2 )}=A(2x + 2 )E [U(Wt+1(t ))] = rf +22A(2x + 2 )= rf +S22AS = /2x + 2 : unconditional Sharpe ratioTimmermann (UCSD) Forecasting Winter, 2017 63 / 64Portfolio Choice under Mean-Variance Utility (cont.)Informed investor knows xt . His stock holdings aret =+ xtA2Et [U(Wt+1(t ))] = rf +(+ xt )22A2Average (unconditional expectation) value of this isE [Et [U(Wt+1(t ))]] = rf +2 + 2x2A2Increase in expected utility due to knowing the predictor variable:E [U inf ] E [Uun inf ] =2x2A2=R22A(1 R2)Plausible empirical numbers, i.e., R2 = 0.005, and A = 3, give anannualized certainty equivalent return of about 1%Timmermann (UCSD) Forecasting Winter, 2017 64 / 64Lecture 2: Univariate Forecasting ModelsUCSD, January 18 2017Allan Timmermann11UC San DiegoTimmermann (UCSD) ARMA Winter, 2017 1 / 591 Introduction to ARMA models2 Covariance Stationarity and Wold Representation Theorem3 Forecasting with ARMA models4 Estimation and Lag Selection for ARMA ModelsChoice of Lag Order5 Random walk model6 Trend and Seasonal ComponentsSeasonal componentsTrended VariablesTimmermann (UCSD) ARMA Winter, 2017 2 / 59Introduction: ARMA modelsWhen building a forecasting model for an economic or financialvariable, the variables own past time series is often the first thingthat comes to mindMany time series are persistentEffect of past and current shocks takes time to evolveAuto Regressive Moving Average (ARMA) modelsWork hors
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