1. Introduction. Hang Qian 1 Iowa State University

Transcription

1 Users Guide to the VARDAS Package Hang Qian 1 Iowa State University 1. Introduction The Vector Autoregression (VAR) model is widely used in macroeconomics. However, macroeconomic data are not always observed at the same frequency. For example, we might have quarterly GDP and monthly CPI data. Shall we average monthly data into quarterly data, or interpolate monthly data from quarterly data? Neither. With the Var(ied) Da(ta) S(ampling), or VARDAS model, we can directly use data of different frequencies as if we were running a standard VAR model for economic forecasting as well as the impulse-response analysis. The VARDAS model is built upon a stationary VAR(p) model with highest frequency data. Complete data are unavailable since some variables might be observed at a lower frequency. Note that lower frequency data (aggregated variables) are mostly formed by linear combinations of high frequency data (disaggregated variables). For instance, the quarterly data are the sum or average of the three monthly data in that quarter. On the one hand, disaggregated variables themselves exhibit auto-correlations and cross-correlations in accordance with the VAR structure. On the other hand, the observed aggregated data impose linear constraints on the interactions of disaggregated variables. That is, these disaggregated variables must sum up to or average to some known numbers, namely the realizations of the aggregated variables. The VARDAS model is estimated in a Bayesian framework, where the latent variables can be conveniently handled by the Gibbs sampler with data augmentation techniques. The basic idea of the VARDAS model is that auto-correlations and cross-correlations make latent disaggregated variables follow a multivariate normal distribution (MVN), while the observed aggregated data further make it a constrained MVN. Once the model parameters are estimated, it is ready for forecasting and studying the dynamic effects of economic shocks as long as suitable identification conditions are imposed. In other words, after the stage of parameter estimation, the VARDAS model is equivalent to a standard reduced form VAR model. The way of recovering structural VAR model from its reduced form remains unchanged. In fact, choosing the VARDAS model instead of aggregating all the data to the lowest frequency has a major advantage on structural VAR identification with zero-contemporaneous-effect constraints. Since the VARDAS model operates on regressions with the (latent) high frequency data, contemporaneity means a shorter time, which effectively relaxes the identification assumption. 1 Contact author: hqi@iastate.edu or matlabist@gmail.com

2 2. Usage The VARDAS package is written in MATLAB, available at Technical details aside, the VARDAS package is friendly to users, who only need to provide the data and the computer will do the rest as routinely as a standard VAR model. The package also comes with graphic users interfaces (GUI) that allow loading the data from an EXCEL file and running the estimation with the mouse clicking. The data can be prepared in an EXCEL file (for GUI users) or stored in a MATLAB float type variable (for MATLAB function users). Suppose we have quarterly GDP and monthly CPI data. In an EXCEL file, prepare your data similar to those in Figure 1. Figure 1: Balanced Aggregation Data in EXCEL DATE GDP CPI Figure 2: Balanced Aggregation Data in MATLAB As is seen in the spreadsheet, the quarterly data should be stored at the last entry of that quarter, namely March, June, September and December. Also make sure that the monthly data starts from the first month of a quarter and ends with the last month of a quarter. Otherwise, the computer has to truncate off the initial and/or last data points. Note that month-quarter aggregation is balanced aggregation, for there are always three months in a quarter. The computer works faster on balanced aggregation, though it does support unbalanced aggregation (see below). Once the computer reads those data into MATLAB, the data variable is shown in Figure 2, where all blank entries are replaced by the data type called. If you prefer to call MATLAB functions to run the VARDAS model, you can prepare your data similar to those in Figure 2. You are recommended to

3 add a transpose sign to your data variable (since during the estimation data are processed in that way), though the computer will do it for you if you do not. Sometimes we need to work on data of unbalanced aggregation. A leading example is that GDP data prior to 1947 does not have quarterly observations. So we encounter quarter-annual aggregation within a time series. In that case, prepare your data in EXCEL similar to those in Figure 3 or in a MATLAB variable similar to Figure 4. Figure 3: Unbalanced Aggregation Data in EXCEL Date GDP 1945Q1 1945Q2 1945Q3 1945Q Q1 1946Q2 1946Q3 1946Q Q Q2 160 Figure 4: Unbalanced Aggregation Data in MATLAB There is no need for the user to specify the aggregation structure, even for the unbalanced aggregation. By counting a run of immediately before a data point reveals the aggregation structure. (That is why an aggregated value is required to be recorded at the end of an aggregation circle.) The user only needs to tell the computer whether the aggregated data is formed by summation or averaging of the disaggregated data. Though the VARDAS package supports data of arbitrary frequencies and unbalanced aggregation in any fashion, the estimation time poses a challenge to a user s patience. The package contains several functions that differ in model generality and estimation speed. VARDAS1: This version only supports two frequencies and balanced aggregation. Some data are of low frequency and some are of high frequency Each series must stick to one frequency and cannot change its frequency halfway. For example: a VAR with monthly GDP and quarterly CPI, M1 etc. VARDAS2: This version only supports two frequencies. A time series changes its frequency once from low to high If more than one series change frequency, the change must occur at the same time For example: A time series with older data being annual and newer being quarterly VARDAS3:

4 This version supports unbalanced aggregation or mixed frequencies in any fashion Disaggregated data are sampled block by block The user specifies the size of the block The sampling speed is slow VARDAS4: This version supports unbalanced aggregation or mixed frequencies in any fashion Disaggregated data are sampled at one time from a giant MVN The sampling speed is very slow Let T be the number of observation, d be the number of component variables in the VAR system, p be the lag order of the VAR, and R be the number of draws in MCMC after the burn-in periods. The input arguments of these functions are Y: Variables in VAR (T-by-d matrix, or d-by-t matrix) nlag: number of lags in the VAR model sum_flag: specify how the latent high frequency data are grouped into low frequency data 1 = sum, 0 = average add_constant: whether to add a constant in the VAR regression 1 = add, 0 = do not add ndraws: number of draws in MCMC burn_in: number of burn-in draws in MCMC The output arguments of these functions are Phi_draws: posterior draws of phi(1),,phi(p), d-by-d-by-p-by-r 4-dimension array Sigma_draws: posterior draws of covariance matrix, d-by-d-by-r 3-dimension array constant_draws:posterior draws of the intercept (not the mean), d-by-r matrix For those who prefer to use graphic interfaces, double click setup.m and press F5 (or type setup in the command window), the GUI will pop up. Load the EXCEL data, click the relevant options, and press the button the estimate the model. In additional, the VARDAS package includes two bonus functions IRF_SHORT.m and IRF_LONG.m to compute the impulse response function of structural shocks. IRF_SHORT.m uses Cholesky decomposition, which effectively imposes one-way recursive restriction of zero contemporaneous effects. On the other hand, IRF_LONG.m identifies structural shocks by long-run neutrality constraints. It should be noted that recovering the structural model from the reduced-form VAR or VARDAS involves scientific insights on the economy and thus had better to be used with care.

5 3. Two Notes Differenced Data The VAR model in use is covariance stationary. However, many macroeconomic variables contain unit roots. It is common to put the first-differenced variables in the VAR system, though in the current model cointegration relations and error correction terms are not included. The VARDAS model with cointegration is left for future research. For differenced data, the aggregated data are not the sum or the simple average of the disaggregated data, but instead a weighed average. To see this, let *Y t + t t=1 be the latent monthly GDP series and we actually put ΔY t = Y t Y t 1 as a component variable in the VAR system. We observe the quarterly GDP series Y t,t+2 = Y t + Y t+1 + Y t+2, t = 1, 4, 7,.. Define the quarterly differenced data Δ 3 Y t,t+2 = Y t,t+2 Y t 3,t 1. The observable quarterly-differenced data and the unobservable monthly-differenced data are linked by the relation Δ 3 Y t,t+2 = 2 j=0 (Y t+j 2 = j=0 (ΔY t+j = ΔY t+2 Y t 3+j ) + 2ΔY t+1 + ΔY t 1+j + ΔY t 2+j ) + 3ΔY t + 2ΔY t 1 + ΔY t 2 To estimate a model with temporal aggregation of differenced data, use VARDAS_DIFF1.m if some data are of low frequency and some are of high frequency; use VARDAS_DIFF2.m if a time series changes its frequency once from low to high. Logarithmic Data Our model is a linear model that handles temporal aggregation by exploring the fact that normality is preserved under linear transformations. Using logarithmic variables in the VAR has many merits, but it also introduces nonlinearity in the aggregation structure. Suppose three monthly data are averaged into a quarterly data such that Y 1,3 = Y 1 + Y 2 + Y 3. If we put logarithmic data in the VAR system, we know (ln Y 1, ln Y 2, ln Y 3 ) are multivariate normal. However, ln Y 1,3 ln Y 1 + ln Y 2 + ln Y 3, which implies that conditional on observed ln Y 1,3 the disaggregated variates do not have a closed form distribution. One way to solve this problem is to redefine the disaggregated data such that ln Y 1,3 = ln Y 1 + ln Y 2 + ln Y 3, where *ln Y t + t t=1 are used as a component series in the VAR system. Note that under this definition the disaggregated data cannot be interpreted as the calendar monthly data. They only bear a statistical interpretation such that the geometric average of the latent ln Y 1, ln Y 2, ln Y 3 equals to the observed ln Y 1,3. The approximation error is almost negligible if monthly changes are small and the geometric averaging works well in practice.

6 4. Technical Details Consider a d-dimension VAR(p) system: Y t = c + Φ 1 Y t Φ p Y t 1 + ε t, where ε t ~N(0, Ω) The reference time unit is t = 1, T, which indexes the highest frequency data in the VAR system. Let Y = (Y t,, Y T ), which is unobservable since some of the component series may be observed at some lower frequencies. We have specified the way constructing the data matrix in Section 2 (Usage). Essentially, we record the aggregated data at the end of the aggregation circle, leaving other entries being. Now we add a transpose sign to this data matrix, and denote it as Y, which is a d-by-t matrix analogue to Y. Suppose Y were known, it was a standard VAR model. The package treats this system as a multi-equation regression model (also called seemingly unrelated regression, SUR), and assigns conjugate, proper and diffused priors to the parameters. Once we set {c, Φ 1,, Φ p } a normal prior and Ω an inverse Wishart prior, the posterior conditionals are normal and inverse Wishart respectively. Now conditional on all the model parameters {c, Φ 1,, Φ p, Ω} as well as our data Y, we discuss how to sample the latent Y. We use the symbol to vectorize a matrix column by column. If the initial Y 1,, Y p come from the stationary distribution of the VAR system, then Y ~N(μ, Γ), where μ = (μ 1,, μ 1 ), μ 1 = (I d Γ 0 Γ 1 Γ T 1 Γ Γ = 1 Γ 0 Γ T 2, ( Γ T 1 Γ T 2 Γ 0 ) p i=1 Φ i where the autocovariance matrix Γ j = E[(Y T μ)(y T j recursively by Γ j = i=1 Φ i Γ j i. p ) 1 c. The dt-by-dt covariance matrix is given by μ) ], j = 0,1,2, can be computed If we knew nothing about Y, the latent Y could be sampled directly from N(μ, Γ). However, our observation on the aggregated variables deepens our understanding on the disaggregated variables. In other words, the observed Y imposes linear constraints on the interaction of Y such that disaggregated variables must sum up to some known numbers, namely those realizations of aggregated data. This motivates us to construct a transformation matrix so as to link Y with Y. To this end, we examine the aggregation structure. Let E be a d-by-t logical matrix according to whether an entry in Y is or not. The transformation matrix A can be constructed in the following way. First, let the

7 main diagonal of A be ones and other elements be zeros. Second, examine each row of the logical matrix E. Suppose we are reading row i and column j of E. If the (i, j) entry is zero, skip and proceed to column j + 1 (or conclude this row). Otherwise, (i, j)entry being one implies a temporally aggregated data is observed. So we search column j 1, j 2, for a run of zeros. Suppose there are M zeros (M 0) in a row immediately before column j; we then add M ones to A. For > 0, the locations in A are row (j 1)d + i, column (j 1)d + i md, m = 1,, M. Note that M = 0 implies one-period trivial aggregation. The new series AY transforms the original series Y in such a way that for a (M + 1)-period temporal aggregation, the first M disaggregated variates are retained, while the last disaggregated variate is replaced by the sum of the disaggregated variates. As a special case, for a one-period aggregation, the variate is simply retained. Clearly, AY can be classified into two blocks: the latent disaggregated variates block and the observed variates block. The latter have their realizations contained in Y. To be exact, AY ~N(Aμ, AΓA ) and Y 0 Y ~N[η 0 + Γ 01 Γ 11 1 (Y 1 η 1 ), Γ 00 Γ 01 Γ 11 1 Γ 10 ], where Y 0 is the subvector of AY selected by the logical vector 1 E, Y 1 is the subvector of Y selected by the logical vector E η 0 and η 1 are two subvector of Aμ selected by the logical vector 1 E and E respectively. Γ 01 is the submatrix of AΓA with rows selected by 1 E and column selected by E Γ 00, Γ 11, Γ 10 are defined similarly. Lastly, Note that in the transformation we squeezed out one disaggregated variate at the end of the aggregation circle, and replaced it with an aggregated data. However, sampling the last disaggregated variates is trivial in that it must equal to the difference between the aggregated value and the sum of the rest disaggregated values. That finishes the cycle of the Gibbs sampler to the VARDAS model. The above sampling procedure is implemented in VARDAS4.m, though the sampling speed is slow due to large matrix manipulation. We can take advantage of the Markov property of the AR process to sample latent variables block by block, which increases the nodes on the MCMC chain but greatly improves the sampling speed. In the package, VARDAS1.m and VARDAS2.m sample latent variables of one aggregation circle in a block, and VARDAS3.m allows the user to choose the block size. Suppose we want to sample Y t,, Y t+j use the joint distribution of Y t p in the block. The idea of the block Gibbs sampler is that we only need to,, Y t,, Y t+j,, Y t+j+p to form the conditional normal distribution due to the Markov property. As for the two ends, we either truncate the variables in use or augment shadow data Y 0, Y 1,. VARDAS1.m and VARDAS2.m use the former method and VARDAS_DIFF1.m and VARDAS_DIFF2.m use the latter.

8 5. Concluding Remarks It is both feasible and sensible to make full use of varied frequency data in a VAR model. The VARDAS model does not involve complicated estimation techniques; it only trivially manipulates normal and conditional normal distributions. From the perspective of a user, all he needs to do is to provide the mixed data and the computer will do the rest. Comments, suggestions and bug reports are more than welcome and appreciated. Reference: Hang Qian, 2011, Vector Autoregression with Varied Frequency Data Full text available at