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9.3.2 Minnesota Prior

Forecasts generated by the unrestricted (so-called reduced-form) VAR in can quickly become imprecise as the dimension of the VAR increases due to the effects of parameter estimation error. This becomes a particular issue in cases where a high lag order, $p,$ is required to account for serial correlation in the variables included in the y vector. If some of the components in y are highly persistent (e.g., interest rates or inflation), while others are not, this could lead to the inclusion of many redundant terms in the model and a large number of unknown parameters. For example, with five variables $(n = 5)$ and four lags $(p = 4)$ , we would need to estimate $n (n p + 1) = 105$ mean parameters.

Allowing the data to inform the model estimates, yet not overwhelm the forecasts due to estimation error, requires a balancing act. To address this issue, Litterman (1979, 1986), and Doan, Litterman, and Sims (1984) introduced what is known as the “Minnesota” priors on the parameters of a VAR. Constructing forecast models with macroeconomic variables modeled in levels, Litterman proceeded to use a random walk as the prior model for each of the equations in the VAR. Hence, the prior for $A_{1}$ is the identity matrix, while for $A_{j}, j > 1$ , the prior is that the coefficients of the matrix are 0. Denoting priors by A, the mean of the prior on the regression coefficients $β$ is specified as

$(\underline{A}_{1})_{ij} = {10 for i = j, for i \neq = j, (\underline{A}_{2})_{ij} = \cdot \cdot \cdot = (\underline{A}_{p})_{ij} = 0.$

The strength of the prior varies with the horizon and also depends on crossvariable effects, with greater shrinkage applied to other variables and their lags.

Specifically, a common variant of the Minnesota prior constructs the prior variance– covariance matrix for the ith equation by setting the variance of the prior on the own lags equal to $π_{1}^{2} / l^{2}$ , where l is the lag length, so the prior gets tighter and tighter as the lag length increases. For lags of other variables in the VAR $(j = 1, \dots, n, j \neq = i)$ , the variance of the prior is set to $π_{1}^{2} π_{2} σ_{i}^{2} / (σ_{i}^{2} l^{2})$ , where $σ_{j}$ is the standard deviation of the errors of the j th equation. The factor $σ_{i} / σ_{j}$ ensures that variables are standardized to have the same scale, as measured by their variance. The hyperparameter π1 measures how tight the prior distribution is around the random walk, with $π_{1} = 0$ denoting the dogmatic random walk prior that disregards all data, while $π_{1} = \infty$ denotes a diffuse prior corresponding to OLS estimation; $π_{2}$ measures the relative tightness of priors on cross-variable effects versus own-variable effects. Finally, $Var (\underline{c}) = \infty$ , corresponding to a diffuse prior for the intercept, i.e., $\underline{c} \sim N (0, κ I)$ , for a very large value of $\dot{θ}_{K}$

In summary, denoting the prior variance of element $i, j$ of $A_{l}$ by $\underline{Var} (A_{l})_{ij}$ we have

$\underline{Var} (A_{l})_{ij} = ⎩ ⎨ ⎧ \frac{π _{1}^{2}}{l ^{2}} \frac{π _{1}^{2} π _{2}}{l ^{2}} (\frac{σ _{i}}{σ _{j}})^{2} for i = j, for i \neq = j . (9.25)$

These define the variance covariance $Ω$ of the prior on $β .$ Finally, for the variance– covariance matrix - it is assumed that the residuals of the VAR equations are contemporaneously uncorrelated with known variances. In practical applications the “known” variances are replaced with their OLS estimates and so $Σ =$ dia $g (\overset{σ}{^}_{1}^{2}, \hdots, \overset{σ}{^}_{n}^{2})$ . Taken together, these are known as the Litterman priors. Because of the independence between parameters in the prior and the diagonal form of $Σ$ the posteriors for each of the equations are also unrelated. Hence the VAR can be estimated equation by equation as in the OLS case.

As noted by Karlsson (2013), the priors $β_{ij} \sim N (\underline{β}_{ij}, τ_{ij}^{2})$ can usefully be thought of in the context of an augmented regression model,

$(r _{j} y _{j}) = (R _{j} Z) β_{j} + (u ~ _{j} u _{j}),$

where $r_{j} = R_{j} β_{j} + \tilde{u}_{j}$ is a set of pseudo data whose ith element is $\underline{β}_{ij} \overset{σ}{^}_{j} / τ_{j}$ , where $\tilde{u}_{j} \sim N (0, \overset{σ}{^}_{i}^{2})$ and the off-diagonal elements of $R_{j}$ are 0, while the diagonal elements are $\overset{σ}{^}_{j} / τ_{ij}$ , for $i, j = 1, \dots, n p$ . Using this representation, the posterior distribution becomes $\overset{ˉ}{β}_{j} ∣ Y_{T} \sim N (\overset{ˉ}{β}_{j}, \overset{ˉ}{Σ}_{j})$ with mean and variance,

$\overset{ˉ}{β}_{j} = (Z^{'} Z + R_{j}^{'} R_{j})^{- 1} (Z^{'} y + R_{j}^{'} r), \overset{ˉ}{Σ}_{j} = \overset{σ}{^}_{j}^{2} (Z^{'} Z + R_{j}^{'} R_{j})^{- 1} .$

In practice, this means that we can use OLS estimation on the augmented regression to compute the posterior mean and variance.

练习题

Which of the following best describes the main issue with unrestricted VAR forecasts as the dimension increases?

A. The forecasts become more accurate due to more data points.

B. The forecasts become imprecise due to parameter estimation error.

C. The forecasts require fewer parameters to estimate.

D. The forecasts become more stable due to reduced lag order.

What is the prior for $A_{1}$ in the Minnesota priors?

A. Zero matrix

B. Diagonal matrix with elements equal to 1

C. Identity matrix

D. Random walk matrix

What is the variance of the prior on the own lags in the Minnesota prior variance-covariance matrix construction?

\frac{π _{1}^{2}}{l}

\frac{π _{1}^{2}}{l ^{2}}

\frac{π _{1}^{2} π _{2}}{l ^{2}}

\frac{π _{1}^{2} π _{2} σ _{i}^{2}}{σ _{j}^{2} l ^{2}}

Which of the following statements about the hyperparameters in the Minnesota prior are correct?

π_{1}

measures the tightness of the prior distribution around the random walk.

π_{2}

measures the relative tightness of priors on cross-variable effects versus own-variable effects.

π_{1} = 0

denotes a diffuse prior corresponding to OLS estimation.

π_{2} = 0

denotes the dogmatic random walk prior that disregards all data.

The strength of the Minnesota prior varies with the horizon and depends on cross-variable effects, with greater shrinkage applied to other variables and their lags.

In the Minnesota prior, the prior variance of element $i, j$ of $A_{l}$ for $i \neq = j$ is $\frac{π _{1}^{2} π _{2}}{l ^{2}} (\frac{σ _{i}}{σ _{j}})^{2}$ .

The mean of the prior on the regression coefficients $β$ for $A_{1}$ is specified as $(A_{1})_{ij} =$ ___ for $i = j$ , and $(A_{1})_{ij} =$ ___ for $i \neq = j$ .

The variance of the prior on lags of other variables in the VAR $(j = 1, \dots, n, j \neq = i)$ is set to $\frac{π _{1}^{2} π _{2} σ _{i}^{2}}{σ _{j}^{2} l ^{2}}$ , where $σ_{j}$ is the standard deviation of the errors of the ___ equation.

Explain the role of $π_{1}$ and $π_{2}$ in the Minnesota prior.

How does the Minnesota prior address the issue of imprecision in unrestricted VAR forecasts?

When using Minnesota priors in a VAR model, what is the primary reason for setting the prior variance of the own lags to $\frac{π _{1}^{2}}{l ^{2}}$ ?

A. To ensure the prior is diffuse for all lag lengths

B. To make the prior tighter as the lag length increases

C. To standardize the variables to have the same scale

D. To disregard all data and rely solely on the prior

In a VAR model with Minnesota priors, the prior for $A_{1}$ is the identity matrix, while for $A_{j}$ ( $j > 1$ ), the prior is that the coefficients of the matrix are 0. This approach helps in reducing the number of parameters to estimate, especially when dealing with highly persistent variables like interest rates or inflation.

The strength of the Minnesota prior varies with the horizon and depends on cross-variable effects, with greater shrinkage applied to other variables and their lags. This is achieved by setting the prior variance of the own lags to $\frac{π _{1}^{2}}{l ^{2}}$ and the variance of the lags of other variables to $\frac{π _{1}^{2} π _{2} σ _{i}^{2}}{( σ _{j}^{2} l ^{2} )}$ , where $σ_{j}$ is the standard deviation of the errors of the $j$ th equation. The factor $\frac{σ _{i}}{σ _{j}}$ ensures that variables are standardized to have the same ___.

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9.3.2 Minnesota Prior