Презентация «Forecasting with bayesian techniques MP»
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Introduction: Two Perspectives in Econometrics Let θ be a vector of parameters to be estimated using data For example, if yt～ i. i. d. N(μ,σ2), then θ=[μ,σ2] are to be estimated from a sample {yt} …
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Outline Why a Bayesian Approach to VARs? Brief Introduction to Bayesian Econometrics Analytical Examples Estimating a distribution mean Linear Regression Analytical priors and posteriors for BVARs …
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Why a Bayesian Approach to VAR? Dimensionality problem with VARs: y contains n variables, p lags in the VAR The number of parameters in c and A is n(1+np), and the number of parameters in Σ is …
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Why a Bayesian Approach to VAR? (2) Usually, only a fraction of estimated coefficients are statistically significant parsimonious modeling should be favored What could we do? Estimate a VAR with …
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Combining information: prior and posterior Bayesian coefficient estimates combine information in the prior with evidence from the data Bayesian estimation captures changes in beliefs about model …
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Shrinkage There are many approaches to reducing over-parameterization in VARs A common idea is shrinkage Incorporating prior information is a way of introducing shrinkage The prior information can be …
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Forecasting Performance of BVAR vs. alternatives Source: Litterman, 1986
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Introduction to Bayesian Econometrics: Objects of Interest Objects of interest: Prior distribution: Likelihood function: - likelihood of data at a given value of θ Joint distribution (of unknown …
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Bayesian Econometrics: Objects of Interest (2) The marginal likelihood… …is independent of the parameters of the model Therefore, we can write the posterior as proportional to prior and data:
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Bayesian Econometrics: maximizing criterion For practical purposes, it is useful to focus on the criterion: Traditionally, priors that let us obtain analytical expressions for the posterior would be …
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Bayesian Econometrics : maximizing criterion (2) Maximizing C() gives the Bayes mode. In some cases (i. e. Normal distributions) this is also the mean and the median The criterion can be generalized …
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Analytical Examples Let’s work on some analytical examples: Sample mean Linear regression model
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Estimating a Sample Mean Let yt～ i. i. d. N(μ,σ2), then the data density function is: where y={y1,…yT} For now: assume variance σ2 is known (certain) Assume the prior distribution of mean μ is …
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Estimating a Sample Mean The posterior of μ: …has the following analytical form with So, we “mix” prior m and the sample average (data) Note: The posterior distribution of μ is also normal: μ～ …
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Estimating a Sample Mean: Example Assume the true distribution is Normal yt~N(3,1) So, μ=3 is known to… God A researcher (one of us) does not know μ for him/her it is a normally distributed random …
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Posterior with prior N(1,1) Compute the posterior distribution as sample size increases
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Posterior with Prior N(1,1/50) Then, we look at more informative (tight) prior and set ν =50 (higher precision)
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Examples: Regression Model I Linear Regression model: where ut～ i. i. d. N(0,σ2) Assume: β is random and unknown but σ2 is fixed and known Convenient matrix representation where The density function …
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Examples: Regression Model I (2) Assume that the prior mean of β has multivariate Normal distribution N(m,σ2M): where the key parameters of the prior distribution are m and M Bayesian rule states: i. …
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Examples: Regression Model I (3) We mix information – densities of data and prior – to get posterior distribution! Result: the density function of β is… … which means that the posterior distribution …
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Since we do not like black boxes… there are 2 ways to get m* and M* (2 parameters to characterize posterior) Since we do not like black boxes… there are 2 ways to get m* and M* (2 parameters to …
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Define a “new” regression model Define a “new” regression model We simply stack our “ingredients” together to mix the information (prior and data) so that now β takes into account both! The GLS …
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Examples: Regression Model II So far the life was easy(-ier), in the linear regression model β was random and unknown, but σ2 was fixed and known What if σ2 is random and unknown?. . Bayesian rule …
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Examples: Regression Model II () To manipulate the product …we assume the following distributions: Normal for data Normal for the prior for β (conditional on σ2): β|σ2 ̴ N(m, σ2M) and Inverse-Gamma …
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Examples: Regression Model II (3) By manipulating the product (see more details in the appendix B) …we get the following result with mean and variance of the posterior for β|σ2 ̴ N(m*, σ2M*) And …
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Priors: summary In the above examples we dealt with 2 types of prior distributions of our parameters: Case 1 prior assumes β is unknown and normally distributed (Gaussian) σ2 is a known parameter the …
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Bayesian VARs Linear Regression examples will help us to deal with our main object – Bayesian VARs A VAR is typically written as where yt contains n variables, the VAR includes p lags, and the data …
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VAR in a matrix form: example Consider, as an example, a VAR for n variables and p=2 Stack the variables and coefficients Then, the VAR Let and rewrite where is a Kroneker product
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How to Estimate a BVAR: Case 1 Prior Consider Case 1 prior for a VAR: coefficients in A are unknown with multivariate Normal prior distribution: and known Σe “Old trick” to get the posterior: use GLS …
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How to Estimate a BVAR: Case 2 (conjugate) Priors Before we see the case of an unknown Σe need to introduce a multivariate distribution to characterize the unknown random error covariance matrix Σe …
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How to Estimate a BVAR: Conjugate Priors Assume Conjugate priors: The VAR parameters A and Σe are both unknown prior for A is multivariate Normal: and for Σe is Inverse Wishart: Follow the analogy …
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BVARs: Minnesota Prior Implementation The Minnesota prior – a particular case of the “Case 1 prior” (unknown model coefficients, but known error variance): Assume random walk is a reasonable model …
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BVARs: Minnesota Prior Implementation The Minnesota prior The prior variance for the coefficient of lag k in equation i for variable j is: … and depends only on three hyperparameters: the tightness …
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BVARs: Minnesota Prior Implementation The Minnesota prior Interpretation: the prior on the first own lag is the prior on the own lag k is the prior std. dev. declines at a rate k, i. e. coefficients …
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Remarks: Remarks: The overall tightness of the prior is governed by γ smaller γ model for yit shrinks towards random walk The effect of other lagged variables is controlled by w smaller estimates …
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BVARs: Prior Selection Minnesota and conjugate priors are useful (e. g. , to obtain closed-form solutions), but can be too restrictive: Independence across equations Symmetry in the prior can …
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Del Negro and Schorfheide (2004): DSGE-VAR Approach Del Negro and Schorfheide (2004) We want to estimate a BVAR model We also have a DSGE model for the same variables It can be solved and linearized: …
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Del Negro and Schorfheide (2004) We estimate the following BVAR: The solution for the DSGE model has a reduced-form VAR representation where θ are deep structural parameters Idea: Combine artificial …
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Del Negro and Schorfheide (2004) Parameter λ is a “weight” of “artificial” (prior) data from DSGE λ=0 delivers OLS-estimated VAR: i. e. DSGE not important Large λ shrinks coefficients towards the …
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Likelihood of the VAR of a DSGE Model Recall the likelihood function for an unconstrained VAR Similarly, the (Quasi-) likelihood for the “artificial” data: which is a prior density for the BVAR …
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Likelihood of the VAR of a DSGE Model
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DSGE-VAR prior
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DSGE-VAR posterior
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Results
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Results
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Kadiyala and Karlsson (1997) Small Model: a bivariate VAR with unemployment and industrial production Sample period: 1964:1 to 1990:4. Estimate the model through 1978:4 Criterion to chose …
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Kadiyala and Karlsson (1997) Compare different priors based on the VAR forecasting performance (RMSE) Standard VAR(p)… … can be rewritten (see slide 29): … and where
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Prior distributions in K&K K&K use a number of competing prior distributions… Minnesota, Normal-Wishart, Normal-Diffuse, Extended Natural Conjugate (see appendix E) … for and Parameters of …
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Prior distributions in K&K
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Forecast Comparison in K&K: Small Model, unemployment
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Forecast Comparison in K&K: Large Model
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Giannone, Lenza and Primiceri (2011) Use three VARs to compare forecasting performance Small VAR: GDP, GDP deflator, Federal Funds rate for the U. S Medium VAR: includes small VAR plus consumption, …
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Giannone, Lenza and Primiceri (2011) The marginal likelihood is obtained by integrating out the parameters of the model: But the prior distribution of is itself a function of the hyperparameters of …
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Giannone, Lenza and Primiceri (2011) We interpret the model as a hierarchical model by replacing pγ(θ)=p(θ|γ) and evaluate the marginal likelihood: The hyperparameters γ are uncertain Informativeness …
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Giannone, Lenza and Primiceri (2011)
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In all cases BVARs demonstrate better forecasting performance vis-à-vis the unrestricted VARs In all cases BVARs demonstrate better forecasting performance vis-à-vis the unrestricted VARs BVARs are …
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Conclusions BVARs is a useful tool to improve forecasts This is not a “black box” posterior distribution parameters are typically functions of prior parameters and data Choice of priors can go: from …
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Thank You! Thank You!
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Appendix A: Remarks about the marginal likelihood Remarks about the marginal likelihood: If we have M1,…. MN competing models, the marginal likelihood of model Mj, f({yt}|Mj) can be seen as: The …
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Appendix A: Remarks about the marginal likelihood Remarks about the marginal likelihood: Predict the first observation using the prior: Record the first observable and its probability, f(y1o). Update …
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Appendix B: Linear Regression with conjugate priors To calculate the posterior distribution for parameters …we assume the following for distributions: Normal for data Normal for the prior for β …
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Appendix C: How to Estimate a BVAR, Case 1 prior Use GLS estimator for the regression Continue (next slide)
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Appendix C: How to Estimate a BVAR, Case 1 Prior Continue So, the moments for the posterior distribution are: The posterior distribution is then multivariate normal
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Appendix D: How to Estimate a BVAR: Conjugate Priors Note that in the case of the Conjugate priors we rely on the following VAR representation … while in the Minnesota priors case we employed Though, …
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Appendix E: Prior and Posterior distributions in Kadiyala and Karlsson (1997)
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Appendix E: Posterior distributions of forecast for unemployment and industrial production in K&K (1997), h=4, T0 =1985:4
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Appendix E: Posterior distribution of the unemployment rate forecast in K&K (1997)
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Appendix E: Choosing λ
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