Silviase/jgov_v1.4_full_ann huggingface dataset:
if the period utility in the habit formation model is given by $ u(C_{t}-\gamma C_{t-1}) $ and the habit parameter $ \gamma $ is equal to $ \omega^{*} $ , the current account in the habit formation model follows the same stochastic process as that in our SI model. $ ^{19} $ In his ICA model with habit formation, Gruber (2004) finds that the model performs better than the model without habits in predicting the current account dynamics. His evidence also suggests good performance of our SI model.

#### 4.2.3 Impulse responses

To better understand the current account in the SI model, it is helpful to investigate the impulse response functions. Figure 2 plots the impulse response functions of $ g_{t} $ , $ ca_{t} $ , and $ \Delta c_{t} $ to one unit increase in $ \varepsilon_{t} $ . The leftmost panel points to the impulse response of $ g_{t} $ . Here we set $ \phi $ at 0.20, so the response of $ g_{t} $ decays quickly. In the middle and right panels, we compare the response of $ ca_{t} $ and $ \Delta c_{t} $ under the RE model ( $ \omega = 0 $ ) and the SI model ( $ \omega = 0.20 $ or 0.80).

Let us first consider the impulse response functions of $ ca_{t} $ and $ \Delta c_{t} $ under the RE model ( $ \omega = 0 $ ). As shown in the blue line in the middle panel of Figure 2, the current account declines in response to $ \varepsilon_{t} $ . Because the shock has a positive permanent effect on endowment, consumers' permanent income increases more than the current endowment. In this case, increases in consumption at the impact period are larger than those in endowment at the same period. As a result, the economy runs current account deficits.

How do the impulse responses differ in the SI model $ (\omega > 0) $ ? When $ \omega = 0.20 $ , the current account is nearly constant, as the coefficient on $ \varepsilon_{t} $ in (14) is close to zero. To perceive the intuition, recall that inattentive consumers let their saving absorb unrecognized shocks between periods of planning. Even though a permanent effect on endowment increases the permanent income of all consumers, some consumers do not reduce their saving. Instead, they unintentionally increase their foreign asset holdings. The unintended increases in foreign asset holdings offset reductions in foreign asset holdings resulting from a permanent shock to endowment.

If the degree of information rigidity is sufficiently high (e.g., $ \omega = 0.8 $ ), the current account increases. In response to a positive shock $ \varepsilon_{t} $ , unintended increases in foreign asset holdings exceed decreases in foreign asset holdings. In this case, the increases in consumption are smaller than those in endowment at the impact period. As a result, the economy runs the current account surplus.
#### 4.2.4 The hybrid SI model

Figure 3 plots the persistence and volatility of the current account and the persistence of changes in consumption against $ \omega $ under the pure and hybrid SI models. In the figure, we set $ \lambda = 0.50 $ for the hybrid SI model. The value is borrowed from the simple average of estimates in Shibata and Shintani (1998) who estimate $ \lambda $ of 11 developed countries.

The proportionality of $ ca_{t}^{HY} $ to $ ca_{t} $ is fully preserved in the hybrid SI model. The relationship of $ \rho_{ca}^{HY} = \rho_{ca} $ continues to hold in the hybrid SI model, and the curve for $ \rho_{ca}^{HY} $ traces out the upward-sloping curve for $ \rho_{ca} $ in the left panel of Figure 3. Therefore, $ \rho_{ca}^{HY} $ can also be high under information rigidity. This result is summarized in the following corollary:

Corollary 4 Under the hybrid SI model with $ \mu=0 $ , the first-order autocorrelation of the current account $ ca_{t}^{HY} $ is given by $ \rho_{ca}^{HY} = (\omega^{*} + \phi) / (1 + \omega^{*}\phi) $ and is the same as that under the pure SI model.

Turning to the volatility of the current account, the proportionality of $ ca_{t}^{HY} $ to $ ca_{t} $ continues to imply that $ V_{ca}^{HY} = (1 - \lambda) V_{ca} $ . The dashed line corresponds to the hybrid SI model with $ \lambda = 0.50 $ , while the solid line points to the pure SI model (i.e., $ \lambda = 0 $ ). The dashed line in the middle panel of Figure 3 indicates that the volatility of the current account in the hybrid SI model is reduced in comparison to the solid line. However, $ V_{ca} $ on the right-hand side can increase with $ \omega $ as long as $ \omega \in (\phi / (1 + r), 1) $ , as in the case of the pure SI model. When $ \phi $ takes a low value and $ \omega $ is sufficiently large, the current account can be volatile in the hybrid SI model. The analytical expression for $ V_{ca}^{HY} $ is provided by the following corollary:

Corollary 5 Under the hybrid SI model with $ \mu=0 $ , the volatility of the current account $ V_{ca}^{HY}=sd\left(ca_{t}^{HY}\right)/sd\left(g_{t}\right) $ is given by

$$ V_{c a}^{H Y}=\left(1-\lambda\right)V_{c a}=\left(1-\lambda\right)\sqrt{\frac{1+\omega^{*}\phi}{\left(1-\omega^{*}\phi\right)\left[1-(\omega^{*})^{2}\right]}}\sqrt{\frac{\left(\omega^{*}-\phi\right)^{2}}{\left(1+r-\phi\right)^{2}}}. $$

How can we describe the persistence of changes in the aggregate consumption in the hybrid SI model? Recall that changes in the aggregate consumption $ \Delta c_{t+1}^{HY} $ is a convex combination of $ \Delta c_{t+1} $ and $ g_{t+1} $ . We demonstrate that persistence of changes in the aggregate consumption is a weighted average of persistence of $ \Delta c_{t} $ and $ g_{t} $ , that is, $ \omega^{*} $ and $ \phi $ . The analytical expression for the persistence of changes in consumption in the hybrid SI model is summarized by the following corollary:
Corollary 6 Under the hybrid SI model with $ \mu=0 $ , the first-order autocorrelation of $ \Delta c_{t} $ is given by the weighted average of $ \omega^{*} $ and $ \phi $ :

$$ \rho_{\Delta c}^{H Y}=\left(1-\Lambda\right)\omega^{*}+\Lambda\phi, $$

where $ \Lambda\in[0,1] $ is a weight given by

$$ \begin{align*}\Lambda\ =&\ \Lambda\left(\lambda,\phi,\omega\right)\\ =&\ \frac{\lambda^{2}Var\left(g_{t}\right)+\lambda\left(1-\lambda\right)Cov\left(g_{t},\Delta c_{t}\right)}{\left(1-\lambda\right)^{2}Var\left(\Delta c_{t}\right)+2\lambda\left(1-\lambda\right)Cov\left(g_{t},\Delta c_{t}\right)+\lambda^{2}Var\left(g_{t}\right)}.\end{align*} $$

Proof: See the Appendix A.5. ☐

The persistence of changes in the aggregate consumption in the hybrid SI model no longer has a one-to-one relationship with $ \omega $ . The extreme right panel of Figure 3 plots the persistence of changes in the aggregate consumption in the pure and the hybrid SI models. When $ \omega $ is low, the dashed line is located above the solid line, increasing the overall persistence of changes in the aggregate consumption by means of the persistence of net output growth. When $ \omega $ is large, the dashed line is located below the solid line, preventing the overall persistence from increasing. $ ^{20} $ In other words, imperfect capital mobility breaks the tight link between the persistence of changes in consumption and the degree of information rigidity.

## 5 Assessment of the SI models

This section assesses the SI models in comparison to the RE models.

### 5.1 Methodology

We begin by slightly generalizing the pure SI model with the possibility of the non-zero mean growth rate of $ g_{t} $ , i.e., $ \mu \neq 0 $ . While this generalization does not affect the steady state of $ \Delta c_{t} $ , the steady state value of $ ca_{t} $ is no longer zero. $ ^{21} $ To address the non-zero steady state, it would be convenient to define $ c\hat{a}_{t} $ and $ \hat{g}_{t} $ by $ c\hat{a}_{t} \equiv ca_{t} - \mathbb{E}(ca_{t}) $ and $ \hat{g}_{t} \equiv g_{t} - \mu $ , respectively. The following proposition generalizes the second part of Propositions 1 and 2:
Proposition 3 Suppose that information rigidity is present as in Propositions 1 and 2. Suppose also that net output growth follows a covariance-stationary $ AR(1) $ process with nonzero mean: $ g_{t} = (1 - \phi)\mu + \phi g_{t-1} + \varepsilon_{t} $ , where $ \varepsilon_{t} \sim i.i.d. $ ( $ 0, \sigma^{2} $ ). Assume that $ \mu $ and $ \omega $ are both sufficiently low such that $ \mu < r $ and $ \omega_{\mu}^{*} \equiv \omega(1 + r) / (1 + \mu) < 1 $ . The stochastic process of $ c\hat{a}_{t} $ and $ \Delta c_{t+1} $ can be approximated by

$$ c\hat{a}_{t}=\left(\omega_{\mu}^{*}+\phi\right)c\hat{a}_{t-1}-\left(\omega_{\mu}^{*}\phi\right)c\hat{a}_{t-2}+\frac{r}{r-\mu}\frac{\omega_{\mu}^{*}-\phi}{\left(1+r\right)/\left(1+\mu\right)-\phi}\varepsilon_{t}, $$

and

$$ \Delta c_{t+1}=\omega_{\mu}^{*}\Delta c_{t}+\left(1-\omega\right)\frac{r}{r-\mu}\frac{\left(1+r\right)\left(1+\mu\right)}{1+r-\phi\left(1+\mu\right)}\varepsilon_{t+1}, $$

respectively.

Proof: See the Appendix A.6. □

Here we impose a restriction that the net output grows at a slower rate $ \mu $ than the foreign asset holdings: $ \mu < r $ . We require this assumption to ensure the stationarity of the non-financial permanent income scaled by the net output. $ ^{22} $ A positive $ \mu $ slightly decreases the persistence of the current account and consumption growth, since a higher $ \mu $ negatively influences $ \omega_{\mu}^{*} $ . Obviously, these results are applicable to the RE models and can be extended to the hybrid SI model.

Before we discuss the performance of the SI models, let us quickly reconfirm the poor performance of the RE models under nonzero $ \mu $ . Table 3 summarizes predictions of the pure and hybrid RE models for the theoretical moments. The left panel reports the predictions of the pure RE model. They are inconsistent with the actual moments (shown in the most right panel) even if nonzero $ \mu $ is taken into account. The predicted persistence of the current account is 0.16 on the cross-country average, which is much lower than the observed persistence of 0.82. The predicted volatility is 0.44 on average, again much lower than the data of 2.29. Of course, changes in consumption have no persistence under $ \lambda = 0 $ .

The poor performance can also be reconfirmed under $ \lambda = 0.5 $ . As we discussed in Section 3, the hybrid RE model can generate serially correlated changes in consumption at the cost of undermining the predicted volatility of the current account. While the persistence of changes in consumption is slightly positive (0.05 on average) due to imperfect capital mobility, the
predicted volatility of the current account is reduced from 0.44 in the pure RE model to 0.22 in the hybrid RE model.

To assess the SI models, we estimate $ \omega $ and $ \lambda $ using the data of 16 developed OECD countries listed in Table 1, rather than using pre-specified values of $ \omega $ and $ \lambda $ . We obtain $ \omega $ and $ \lambda $ to match the theoretical moments of the persistence and volatility of the current account and the persistence of changes in consumption with the sample moments in each country. Let $ f(\theta) $ be a vector of distances between the theoretical and sample moments for the persistence and volatility of the current account and the persistence of changes in consumption, where $ \theta $ denotes the vector of parameter consisting of $ \omega $ and $ \lambda $ . In the case of the hybrid SI model, for example, $ f(\theta) = \left[\rho_{ca}^{HY}(\theta) - \rho_{ca}^{data}, V_{ca}^{HY}(\theta) - V_{ca}^{data}, \rho_{\Delta c}^{HY}(\theta) - \rho_{\Delta c}^{data}\right]' $ . Using the data of $ ca_t $ and $ \Delta c_t $ in each country, we obtain the weighting matrix $ \hat{W} $ from the inverse of the bootstrap covariance matrix of $ \left(\rho_{ca}^{data}, V_{ca}^{data}, \rho_{\Delta c}^{data}\right)' $ . The objective function $ Q(\theta) $ to be minimized is defined by the quadratic form:

$$ Q\left(\theta\right)=f\left(\theta\right)^{\prime}\hat{W}f\left(\theta\right). $$

When minimizing $ Q \left( \theta \right) $ , we impose the restriction that $ \omega \in \left[ 0, \left( 1 + \mu \right) / \left( 1 + r \right) \right) $ to ensure that $ \omega_{\mu}^{*} < 1 $ for both of the pure and hybrid SI models. We further impose the restriction that $ \lambda = 0 $ in the pure SI model and $ \lambda \in [0, 1) $ in the hybrid SI model, respectively. The minimizers of $ Q \left( \theta \right) $ are denoted by $ \tilde{\theta} = \left( \tilde{\omega}, 0 \right)^{\prime} $ for the former and $ \hat{\theta} = \left( \hat{\omega}, \hat{\lambda} \right)^{\prime} $ for the latter, respectively.

### 5.2 Predictions of the SI models

The left panel of Table 4 shows the predictions of the pure SI model and the middle panel presents those of the hybrid SI model. The two models' predictions are based on our estimates of $ \omega $ and $ \lambda $ , which we will discuss in the next section. For comparison, the right panel again reports the actual moments from the OECD countries.

Let us first consider the pure SI model. Overall, it exhibits a great improvement in comparison to the RE models in Table 3. For example, the pure SI model can almost fully account for the three targeted moments in the Netherlands. In particular, the predicted moments are $ (\rho_{ca}, V_{ca}, \rho_{\Delta c}) = (0.83, 2.05, 0.76) $ , as opposed to the actual moments: $ (\rho_{ca}^{data}, V_{ca}^{data}, \rho_{\Delta c}^{data}) = (0.85, 2.07, 0.74) $ . The SI model also performs well in other countries, though all moments are not fully explained. For example, the predicted moments in Den-
mark are $ (\rho_{ca}, V_{ca}, \rho_{\Delta c}) = (0.77, 1.75, 0.78) $ , while the actual moments are $ (0.94, 3.73, 0.20) $ . This outcome is a remarkable improvement in predictions relative to the RE models, because the RE models predict $ (\rho_{ca}, V_{ca}, \rho_{\Delta c}) = (-0.01, 0.02, 0.00) $ and $ (\rho_{ca}^{HY}, V_{ca}^{HY}, \rho_{\Delta c}^{HY}) = (-0.01, 0.01, 0.01) $ .

While the pure SI model remarkably improves predictions of the current account, it tends to overpredict the persistence of changes in consumption. In particular, $ \rho_{\Delta c} $ is 0.65 in the cross-country average but the actual persistence is only 0.25, which is much lower than in the model. The overprediction stems from the one-to-one relationship of $ \rho_{\Delta c} = \omega_{\mu}^{*} = \omega(1 + r) / (1 + \mu) $ in (23). That is, although a large $ \omega $ helps the SI model explain the persistence and volatility of the current account, the one-to-one relationship between $ \rho_{\Delta c} $ and $ \omega $ also increases the persistence of changes in consumption far beyond the observed persistence.

The hybrid SI model breaks the tight link. The middle panel of Table 4 presents the theoretical moments generated by the hybrid SI model. The hybrid SI model continues to predict a persistent and volatile current account but now with a moderate degree of persistent consumption growth. In the example of Denmark, the prediction of the hybrid SI model is $ \left(\rho_{ca}^{HY}, V_{ca}^{HY}, \rho_{\Delta c}^{HY}\right) = (0.98, 3.83, 0.21) $ as opposed to the data of $ (0.94, 3.73, 0.20) $ . In terms of the cross-country average, the predicted persistence and volatility of the current account is 0.94 and 2.39, respectively. Unlike that in the pure SI model, the predicted persistence of changes in consumption is now suppressed to 0.34, which is fairly close to the data of 0.25.

It may not be surprising that imperfect capital mobility leads to the good performance of the SI model because the hybrid SI model has one additional free parameter $ \lambda $ . However, imperfect capital mobility alone does not necessarily improve the predictions, unless information rigidity is incorporated. When we estimate $ \lambda $ in the hybrid RE model with the same objective function, the estimated $ \lambda $ turns out to be zero in 11 out of 16 countries. $ ^{24} $ In our exercise, the improvement of predictions are achieved only under the information rigidity.

### 5.3 Degrees of information rigidity and imperfect capital mobility

While the hybrid SI model achieves a fairly good model performance, we should note a caveat in the estimated values of $ \omega $ and $ \lambda $ . The hybrid SI model can achieve good performance under relatively high degrees of information rigidity and imperfect capital mobility. Table 5 shows $ \omega $ and $ \lambda $ that are estimated for each country in the two SI models. On average, $ \hat{\omega} $ is 0.64.
in the pure SI model and 0.89 in the hybrid SI model. These values imply that the average duration of holding information until the next update is 2.8 years in the pure SI model and is 9.1 years in the hybrid SI model. $ ^{25} $ The latter value seems extremely large compared to the average duration estimate of 1.3 years obtained by Mankiw and Reis (2007).

We also note that an improvement with the hybrid SI model relies on a high degree of imperfect capital mobility in some countries. While $ \hat{\lambda} $ is only 0.08 in the Netherlands, it is almost one in Belgium and Germany under the parameter restriction of $ \lambda \in [0,1) $ . Overall, however, the cross-country average of $ \hat{\lambda} $ is 0.58. This average is not extremely high in the consumption literature though it may be considered to be high compared to the estimates of some previous studies. $ ^{26} $ For single country's estimates, they are statistically significant in many countries. In the right panel of Table 5, we report $ Q(\tilde{\theta}) - Q(\hat{\theta}) $ , the difference between the objective functions evaluated at $ \tilde{\theta} $ and $ \hat{\theta} $ . The difference is asymptotically distributed as $ \chi^{2}(1) $ under the null hypothesis of $ \lambda = 0 $ . In 11 out of 16 countries, $ \lambda $ is found to be statistically significant at the conventional significance level.

An interpretation of the high degrees of information rigidity and imperfect capital mobility may be that the ICA model does not allow for other possible frictions and that the estimated parameters may reflect such frictions outside the model. For example, we noted that the current account in the SI model is observational equivalent to that in the habit formation model. Since our assessment of the SI models does not allow for habit formation, a high degree of information rigidity may, in fact, reflect the habit formation in some countries. Therefore, while the SI model continues to be a promising method in explaining the current account, introducing other types of frictions and/or shocks into SI model may result in more reasonable estimates for the degrees of information rigidity and imperfect capital mobility.

## 6 Conclusions

This paper extends the intertemporal current account (ICA) model with sticky information (SI). In our ICA model, consumers are inattentive to shock to net output and infrequently update their information, as developed by Mankiw and Reis (2002, 2007) and Reis (2006). Under the assumption that a permanent shock to net output drives the current account, the
rational expectations model fails to predict a persistent and volatile current account as well as persistent changes in consumption. In our SI models, information rigidity improves the predicted persistence and volatility of the current account. If the SI model is extended with imperfect capital mobility as in Shibata and Shintani (1998), our SI model exhibits a fairly good performance in predicting changes in consumption as well as the current account. In particular, the SI model almost perfectly explains the data of the 16 OECD countries if the degrees of information rigidities and imperfect capital mobility are sufficiently high.

Our analysis also suggests that these frictions may need to be introduced into a richer model of the current account to reduce too much reliance on imperfect information and imperfect capital mobility. Toward this end, it may be important to consider the production economy and to include other frictions in the ICA model. $ ^{27} $ Consideration of other shocks, such as the world interest rate shock and the exchange rate shock, may also be helpful. $ ^{28} $ Extending the ICA model in these directions under information rigidity would be an important step for future research.

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