Factor Analysis II Dynamic Factor Models

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  • Summit Consulting, LLC

    2013 Summit Consulting, LLC

    This presentation is property of Summit Consulting and is not to be reused or resdistributed without permission. www.summitllc.us.

    Factor Analysis II: Dynamic Factor Models

  • 2

    Introduction: Recall (Static) Factor Analysis

    Factor Analysis II: Dynamic Factor Models

    Model: =+

    Extraction methods: Principal components: Find the linear combination that explains the maximum

    variance from the Xs. This is the first factor. Then find the next combination that explains the maximum proportion of the remaining variance and is orthogonal to the next factor, etc. (proceed until all variance is explained)

    Maximum Likelihood Estimation: Find the factors that maximize correlation with variables; need to assume something about the distribution (usually multivariate normal)

    Estimation: Predict factors using weighted sum (F = X) or Regression Scoring (F = X XX 1 )

    Use predicted factors in original regression

    2013 Summit Consulting, LLC

  • 3

    Extension to Dynamic Setting

    Factor Analysis II: Dynamic Factor Models

    This is all for a static case (i.e. we are not considering time dynamics)

    But what if we have data collected over time?

    Issue: Data can have hundreds of series, but number of time observations on each series relatively short. Big N, small T issue

    Solution: (Dynamic) Factor Analysis Sargent and Sims (1977): two dynamic factors can explain a large fraction of variance

    in important U.S. quarterly macroeconomic variables.

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  • 4

    Dynamic Factor Model

    Factor Analysis II: Dynamic Factor Models

    Suppose we want to forecast a variable, yt, using observations Xt, their lags, and lags of yt. The issue, however, is the dimensionality problem discussed above.

    Idea of DFM same as in static case: few factors ft drive co-movements of Xt, which may have large dimensionality.

    Model: Xt = L ft + et

    ft= L ft1 + t

    Assumptions: E ettk

    = 0 for all k

    E eitejs = 0 for all s if i j

    Similar to in static case, i(L) is the dynamic factor loading for the ithseries

    Xit.

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  • 5

    Benefits of DFM

    Factor Analysis II: Dynamic Factor Models

    If we know the factors and the distribution of ( , ) then it turns out we can make efficient forecasts for using a regression of on lagged factors and lags of .

    We avoid the curse of dimensionality because we can capture the effect of N variables with only q factors, with q

  • 6

    Estimating Factors in DFM

    Factor Analysis II: Dynamic Factor Models

    Three approaches: 1. Use MLE and the Kalman Filter

    2. Use Principal Components Extraction

    3. Use a combination of the two

    Can think of these as three generations of approaches, that have been refined over time.

    The pre-historic DFMs of Sargent and Sims (1977) did not estimate factors (just looked for evidence of dynamic factor structure and estimate the importance of the factor).

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  • 7

    First Approach: MLE and the Kalman Filter

    Factor Analysis II: Dynamic Factor Models

    Idea: Use the Kalman filter to construct the likelihood function, estimate the parameters by MLE, and then use the KF and smoother to obtain efficient estimates of factors. KF in a nutshell: a linear algorithm that basically uses an updating technique every

    period to construct efficient forecasts. Useful when you have unobserved series.

    Model is a linear state space model: = +

    =

    =

    Here = (, 1 , ,

    )

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  • 8

    Estimation in the First Approach

    Factor Analysis II: Dynamic Factor Models

    With some standard assumptions, the model above defines a complete linear state space model. Can use the KF to compute the likelihood, use MLE to estimate the parameters and then, given the parameters, use the KF and smoother to estimate filtered values of

    Advantage: This state space formulation can handle data irregularities (for example, if some series are observed weekly and others monthly)

    Ex: Angelini, Banbura and Runstler (2008) use a DFM-model with mixed monthly and quarterly data to forecast monthly GDP in Euro-area

    Disadvantage: The number of parameters will be proportional to N. Since we usually have a large N in the first place, this method can be cumbersome in most desired applications involving large systems.

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  • 9

    Second Approach: Principal Components

    Factor Analysis II: Dynamic Factor Models

    Solution to a least-squares problem:

    min1,,,

    1

    ( )( )=1

    This minimization problem turns out to be equivalent to: max

    s. t. N

    1 = Ir

    The solution to this problem is to set equal to the eigenvectors of corresponding to the r largest eigenvalues.

    Then it turns out we can predict the factors as:

    = 1

    This is just like the principal components estimator in the static case.

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  • 10

    Estimation in the Second Approach

    Factor Analysis II: Dynamic Factor Models

    The principal components estimator of is consistent (Stock and Watson, 2002).

    Since in time-series/panel data there is always the potential of heteroskedasticity/autocorrelation, one can use a generalized principal components estimator (similar to GLS).

    The issue with this estimator is that it only is averaging across series, not across time. That is, you basically need to do extraction every period, but the method does not account for potential dynamics of the series.

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  • 11

    Third Approach: Combining Principal Components and State Space Methods

    Factor Analysis II: Dynamic Factor Models

    Merge the statistical efficiency of the state space approach with the convenience of the principal components approach.

    Use of the KF allows average across series and time, not just across series as in PC estimators.

    In this way, we can get improvements in the estimates of the factors.

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  • 12

    Estimation in Third Approach

    Factor Analysis II: Dynamic Factor Models

    Two step procedure: Giannone, Reichlin, and Small (2008); Doz, Giannone and Reichlin (2006). 1. Estimate factors using principal components (or GPC)

    2. Use the estimated factors to estimate the unknown parameters of the state space representation.

    Second step depends on whether state vector is specified in terms of the static or dynamic factors.

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  • 13

    Parameter Estimation in Third Approach: Static Factors

    Factor Analysis II: Dynamic Factor Models

    State space model given by: = +

    =

    =

    Given , estimate by regressing X on , use residuals to estimate univariate autoregressions.

    Also estimate by regressing on lags and use those residuals to estimate .

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  • 14

    Parameter Estimation in Third Approach: Dynamic Factors

    Factor Analysis II: Dynamic Factor Models

    State space model given by: = +

    = 1 +

    =

    Note the difference here is that and its lags explicitly enter the state vector.

    Given estimates one can estimate via a regression of on and the coefficients via the residuals from that regression.

    Can also estimate by estimating a VAR for .

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  • 15

    Estimation in Third Approach

    Factor Analysis II: Dynamic Factor Models

    Given that we have these estimated parameters, one can then compute an improved estimate of or (which now invokes time-series averaging) using the KF and smoother. Note: can also use these estimated coefficients as starting values for MLE

    estimation of the coefficients.

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  • 16

    Determining the Number of Factors

    Factor Analysis II: Dynamic Factor Models

    Static factors: Use scree plot/information criteria (for example, Bai and Ng (2002)). Basically use a penalized version of MLE/Principal Components where you compute

    the trade-off between the benefit of including an additional factor against the cost of increased sampling variability from estimating another parameter.

    Dynamic factors: Hallin and Liska (2007)

    Bai and Ng (2007)

    Amenguel and Watson (2007)

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  • 17

    Uses of Estimated Factors (Estimated Using Any of the Three Approaches)

    Factor Analysis II: Dynamic Factor Models

    Use in second stage regressions: Forecasting: Can compute one step ahead and multistep forecasts of using

    factors and their lags.

    For multi-step case, can either regress + on , and their lags (direct method), or can first estimate a VAR for , and then use this VAR in conjunction with the one-step ahead forecasting equation to iterate forward h periods (iterated method).

    Either direct or iterated method can be better; mixed evidence on this.

    Factors as instrumental variables

    Kapetanios and Marcellino (2008), Bai and Ng (2010)

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  • 18

    Stata Application

    Factor Analysis II: Dynamic Factor Models

    Stata command: dfactor New i