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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} Classical perspective: there is an unknown true value for θ we obtain a point estimator as a function of the data: Bayesian perspective: θ is an unknown random variable, for which we have initial uncertain beliefs - prior prob. distribution we describe (changing) beliefs about θ in terms of probability distribution (not as a point estimator!)

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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 Prior selection in applications (incl. DSGE-VARs)

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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 n(n+1)/2 Assume n=4, p=4, then we are estimating 78 parameters, with n=8, p=4, we have 133 parameters A tension: better in-sample fit – worse forecasting performance Sims (Econometrica, 1980) acknowledged the problem: “Even with a small system like those here, forecasting, especially over relatively long horizons, would probably benefit substantially from use of Bayesian methods or other mean-square-error shrinking devices…”

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$Why a Bayesian Approach to VAR? (2) Usually, only a fraction of estimated coefficients are statistically significant parsimonious modeling should be favored What could we do? Estimate a VAR with classical methods and use standard tests to exclude variables (i.e. reduce number of lags) Use Bayesian approach to VAR which allows for: interaction between variables flexible specification of the likelihood of such interaction$

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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 classical methods and use standard tests to exclude variables (i.e. reduce number of lags) Use Bayesian approach to VAR which allows for: interaction between variables flexible specification of the likelihood of such interaction

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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 parameters Prior: initial beliefs (e.g., before we saw data) Posterior: new beliefs = evidence from data + initial beliefs

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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 reduced to a few parameters, i.e. hyperparameters

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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 parameters and observables/data): Marginal likelihood: Posterior distribution: i.e. what we learned about the parameters (1) having prior and (2) observing the data

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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 needed Today, with increased computer power, we can use any prior and likelihood distribution, as long as we can evaluate them numerically Then we can use Markov Chain Monte-Carlo (MCMC) methods to simulate the posterior distribution (not covered in this lecture)

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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 to: λ controls relative importance of prior information vs. data

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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 normal, μ～ N(m,σ2/ν): where the key parameters of the prior distribution are m and ν

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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～ N(m*,σ2/{ν+T}) Diffuse prior: ν→0 (prior is not informative, everything is in data) Tight prior: ν→ ∞ (data not important, prior is rather informative)

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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 variable μ~N(m,1/v) The researcher initially believes that m=1 and ν=1, so his/her prior is μ~N(1,1)

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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 for data is:

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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.e., the posterior of β is proportional to the product of the data density of data and prior

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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 is again (!) normal with the mean and variance

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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 characterize posterior) The long: manipulate the product of density functions (see Hamilton book, p367) The smart: use GLS regression… We have 2 ingredients: prior distribution , which implies and our regression model that “catches” the impact of the data on the estimate of β

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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 estimator of β… is exactly our posterior mean And the posterior variance of β is

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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 states: i.e., the posterior of β and σ2 is proportional to the product of the density of data, prior of β (given σ2) and prior of σ2

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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 for the prior for σ2 : σ2 ̴ IG(λ,l) Note: inverse-gamma is handy! It guaranties that random draws σ2 >0!

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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 parameters for posterior for σ2 ̴ IG(λ*,l*)

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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 assumption Gaussian errors delivers posterior normal distribution for β Case 2 (conjugate) priors assumes β and σ2 are unknown β and σ2 have prior normal and Inverse-Gamma distributions respectively with Gaussian errors delivers posterior distributions for β and σ2 of the same family

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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 sample size is T We have seen that it is convenient to work with a matrix representation for a regression Can we get it for our VAR? Yes! …and it will help to get posteriors for our parameters

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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 estimator (appendix C for details) Result So the posterior distribution is multivariate normal

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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 Consider a matrix Each raw is a draw form N(0,S) The nxn matrix has an Inverse Wishart distribution with k degrees of freedom: Σe~IWn(S,l) If Σe ~ IWn(S,l), then Σe-1 follows a Wishart distribution: Σe-1~Wn(S-1,l) Wishart distribution might be more convenient Σe-1 is a measure of precision (since Σe is a measure of dispersion)

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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 with univariate regression examples to put down the moments for posterior distributions Recall matrix representation for our VAR: Posterior for A is multivariate normal: Posterior for Σe is Inv. Wishart: See appendix D for details

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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 for every yit in the VAR Hence, for every yit coefficient for the first own lag yit-1 has a prior mean of 1 coefficients for all other lags yit-k , yjt-1 , yjt-k have 0 prior mean So, our prior for coefficients of VAR(2) example would be:

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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 parameter γ (typically the same in all equations) and the relative weight parameter w: is 1 for own lags and <1 for other variables parameter q governs the tightness of the prior depending on the lag (often set to 1) is a “scale correction” the ratio of residual variances for OLS-estimated AR:

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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 for longer lags are more likely to be close to 0 the prior on the first lag of another variable is the prior std. dev. is reduced by a factor w: i.e. it is more likely that the first lags of other variables are irrelevant the prior std. dev. on other variables’ lags declines at a rate k

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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 shrink towards AR model (yit is not affected by yjt) Practitioner’s advice (RATS Manual) on the choice of hyperparameters: Set γ=0.2, =0.5 Focus on forecast errors statistics, when selecting alternative hyperparameters Loosen priors on own lags and tighten on other lags to improve Substitute priors manually if there is a strong reason

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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 sometimes be a problem Increased computer power allows to simulate more general prior distributions using numerical methods Three examples: DSGE-VAR approach: Del Negro and Schorfheide (IER, 2004) Explore different prior distributions and hyperparameters: Kadiyala and Karlsson (1997) Choosing the hyperparameters to maximize the marginal likelihood: Giannone, Lenza and Primiceri (2011)

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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: approximated with a RF VAR Then, we can use coefficients from the DSGE-based VAR as prior means to estimate the BVAR Several advantages: DSGE-VAR may improve forecasts by restricting parameter values At the same time, can improve empirical performance of DSGE relaxing its restrictions Our priors (from DSGE) are based on deep structural parameters consistent with economic theory

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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 and T actual observations (Y,X) and to get the posterior distribution T*=λT “artificial” observations are generated from the DSGE model: (Y*,X*)

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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 DSGE solution: i.e. data not important to find an “optimal” λ marginal likelihood is maximized (appendix E) Can implement the procedure analytically… let’s see

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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 parameters Rewrite the likelihood for the “artificial” data (open brackets)

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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 hyperparameters: forecasting performance over 1979:1-1982:3 Use the remaining sub-sample 1982:4-1990:4 for forecasting Large “”Litterman” Model: a VAR with 7 variables (real GNP, inflation, unemployment, money, investment, interest rate and inventories) Sample period: 1948:1 to 1986:4. Estimate the model through 1980:1 Use the remaining sub-sample 1980:2-1986:4 for forecasting

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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 the prior distribution for : each yit is a random walk (just as in Minnesota priors above) The variance of each coefficient depends on two hyper-parameters w, :

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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, investment, hours worked and wages Large VAR: expand the medium VAR with up to 22 variables The prior distributions of the VAR parameters ϴ={, Σ, Σe} depend on a small number of hyperparameters The hyperparameters are themselves uncertain and follow either gamma or inverse gamma distributions This is to the contrast of Minnesota priors where hyperparameters are fixed!

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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 the model i.e. p(θ)=p (θ|γ)

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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 of their prior distribution is chosen via maximizing the posterior distribution Maximizing the posterior of γ corresponds to maximizing the one-step ahead forecasting accuracy of the model

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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 roughly at par with the factor models, known to be good forecasting devices

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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 a simple Minnesota prior (that is convenient for analytical results) …to a full-fledged DSGE model that incorporates theory-consistent structural parameters The choice of hyperparameters for the prior depends on the nature of the time series we want to forecast No “one size fits all approach”

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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 update on the weight of model Mj after observing the data The out-of-sample prediction record of model j. Model comparison between two models is performed with the posterior odds ratio: Favor’s parsimonious modeling: in-built “Occam’s Razor.”

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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 your beliefs: Predict the second observation: Record f(y2o|y1o). Eventually, you get f({yo})=f(y1o) f(y2o|y1o)…..f(yTo|y1o, y2o,…, yT-1o).

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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 (conditional on σ2): β|σ2 ̴ N(m, σ2M) and Inverse-gamma for the prior for σ2 : σ2 ̴ IΓ(λ,k) Next consider the product

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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, if we have priors for vectorized coefficients in the form we can also get priors for coefficients in the matrix form For the mean we simply need to convert α back to the matrix form A The variance matrix for can be obtained from the variance for :