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# Experience-based forecasts aggregate to average forecasts

2021-07-16 来源: 51Due教员组 类别: Essay范文

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**Experience-based forecasts aggregate to average forecasts**

Abstract

Our analysis so far has focused on explaining heterogeneity in expectations and financial decisions based on heterogeneity in inflation experiences. We now test whether learning from experience also helps explain aggregate dynamics. We show that experience-based forecasts aggregate to average forecasts that closely resemble those from constant-gain algorithms in the existing literature, which have been shown to explain macroeconomic time series data. We argue that learning from experience provides a micro-underpinning for adaptive-learning models, but offers conceptual and econometric advantages in the identification of the structural parameters that pin down the learning rule.

Explaining aggregate expectations

We now test directly how well the learning-from-experience model matches aggregate survey expectations. Figure 7 shows both the time path of averages from the raw survey data (circles) and average experience-based forecasts (solid line), as before based on θ = 3.044. Since our imputation of percentage responses only targeted cross-sectional differences, but not the average level of percentage expectations, we omit all periods in which we only have categorical inflation expectations data.

It is apparent from the figure that the average learning-from-experience forecasts track the average survey expectations closely. The good match is by no means mechanical: Our estimation of θ used only cross-sectional differences in survey expectations, but no information about the level of the average survey expectation. It could have been possible that the θ that fits cross-sectional differences produces a time path for average expectations that fails to match average survey expectations. As the figure shows, though, the two time paths match well. Figure 7 also shows the time path of constant-gain-learning forecasts, using γ = 0.0180 from Figure 6. Not surprisingly, given that γ was chosen to minimize the distance in the implied weights, the forecasts are almost indistinguishable.

This illustrates further that, at the aggregate level, the learning-from-experience expectations formation mechanism can be approximated well with constant-gain learning. Finally, we compare the average learning-from-experience forecast to a sticky-information forecast. Sticky information, as in Mankiw and Reis (2002) and Carroll (2003), induces stickiness in expectations, and it is possible that our estimation of the learning-from-experience rule might be picking up some of this stickiness in expectations.

We evaluate the economic and statistical significance of this graphical impression in Table 4. We regress the average survey expectations in quarter t on the average forecast predicted by learning-from experience (column (i)), by constant-gain learning (column (ii)), and by the sticky-information model (column (iii)). The learning-from-experience model in (10) predicts a coefficient on the experience-based forecast of one, and column (i) shows that the estimated coefficient of 0.887 is close to one, and less than one standard error away from it. With 56.4% the adjusted R2 is high. This confirms the informal graphical impression in Figure 7 that the learning-from-experience forecast closely tracks the actual average survey expectations. Not surprisingly, given the similarity of ¯τt+1|t and constant-gain learning forecasts using γ = 0.0180, the constant-gain learning forecast produces almost identical results. The explanatory power of the sticky-information forecast in column (iii) is also similar, only a bit lower and a bit noisier, and the adjusted R2 is slightly higher. Most importantly, if we add the sticky-information forecast as an explanatory variable along with the learning-fromexperience forecast (column (iv)), the coefficient on the learning-from-experience forecast becomes slightly smaller, but remains large (also relative to the sticky-information coeffi- cient) and significant. Hence, we can conclude that the learning-from-experience forecast does not just pick up the sticky-information effect of Mankiw and Reis (2002) and Carroll (2003).

A foundation of perpetual learning Perpetual learning plays a centrol role in explaining macroeconomic dynamics, as emphasized, for example, by Sargent (1999), Orphanides and Williams (2005a), and Milani (2007). It is therefore important to identify the underlying reasons for perpetual learning; only then is it possible to predict the circumstances under which economic agents update with a high or low gain. Despite their similarity at the aggregate level, models of experience-based learning and constant-gain learning differ fundamentally in their motivation for the down-weighting of past data, and resulting perpetual learning. The standard motivation in constant-gain models for the discounting of past data, and resulting perpetual learning, is the concern that structural changes or drifting parameters have rendered historical data from the distant past irrelevant for the estimation of current parameters of the perceived law of motion. Learning from experience attributes the down-weighting to memory of past data being lost as older generations 36 die and new ones are born.

Out-of-sample predictions

Another advantage of the learning-from-experience model is that it makes predictions about cross-sectional heterogeneity in expectations. This in turn implies that empirically observed cross-sectional heterogeneity in expectations provides useful information that can 39 help estimate the parameters of individuals’ learning rules. This is a key difference from representative-agent applications of adaptive learning models. As Chevillon, Massmann, and Mavroeidis (2010) show, the identification of structural parameters in representative-agent macro models with adaptive learning is difficult, and the problems are magnified if the parameters of the learning rule are unknown and need to be estimated.

Fitting the learning rule to the time path of mean or median survey expectations can help pin down the learning-rule parameters, but the estimates may be imprecise. Within the learning-from-experience model, the gain parameter θ can be identified from cross-sectional data. This brings in a new dimension of data that can help pin down the learning dynamics. We illustrate this point by comparing the out-of-sample fit of the different models. In our estimations so far, we have shown that we obtain the most precise estimates of the gain parameter θ when we focus purely on cross-sectional variation by employing time dummies (Table 1). Here we show that this way of estimating the gain also yields the best pseudo-out-of-sample fit to the time-series of mean survey expectations.

Figure 9 compares the pseudo-out-of-sample fit of the learning-from-experience model with the constant-gain learning model. We estimate the gain parameters in both models recursively, with expanding windows, where the first window extends from 1953Q4 to 1977Q4. For each window, expectations data until quarter t − 1 is used to estimate the gain parameter (mean expectations in case of the constant-gain model, and cohort data as in column (ii) of Table 1 in case of the learning-from-experience model), and we then predict, based on this gain estimate and historical inflation data until t − 1, the mean inflation survey expectation in quarter t. In case of the learning-from-experience model this prediction is given by ¯τt+1|t as in (10), but with θ estimated only from expectations data up to quarter t − 1; in the case of constantgain learning it is simply the fitted value of the constant-gain learning rule. The figure plots the cumulative sum of squared errors from these predictions from 1978Q1 to the end of the sample

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