Silviase/jgov_v1.4_full_ann huggingface dataset:
normalization is useful in describing the current account dynamics, particularly when the growth rate of net output is stationary. A similar normalization was first employed by Campbell and Deaton (1989) in the context of the optimal saving rate under the permanent income hypothesis when labor income growth is stationary. For the same reason, we also express changes in consumption as a fraction of lagged net output. The same reformulation of consumption change has also been used by Shibata and Shintani (1998) in their analysis of ICA model with limited capital mobility.

Table 1 assesses current account dynamics in the sample countries. In the table, we report the cross-country average of persistence and volatility of the current account, along with the persistence of net output growth and changes in consumption. The persistence of the current account is measured by the first-order autocorrelation, denoted by $ \rho_{ca}^{data} $ . $ ^{12} $ Volatility is measured by the standard deviation ratio of the current account to net output growth. We denote the volatility measure by $ V_{ca}^{data} $ . In this measure, when the current account is more volatile than the net output growth, $ V_{ca}^{data} $ exceeds one.

Table 1 indicates that the current account is much more persistent and volatile than the net output growth. We see that the observed persistence of the current account $ \left(\rho_{ca}^{data}\right) $ is 0.82, which is much more persistent than that of the net output growth of 0.16 $ \left(\phi^{data}\right) $ . While the table reports, for brevity, only the persistence averaged across countries, these inequalities are preserved for all single-country data. When we examine the volatility, the current account is about twice as volatile as the net output growth. In terms of the single-country data, the volatility of the current account in all countries except for Japan exceeds unity. We note that the persistence of changes in consumption $ \left(\rho_{\Delta c}^{data}\right) $ deviates from zero, and is 0.25 on average.

## 3 The RE models

In this section, we present RE models, focusing on the persistence and volatility of the current account.

### 3.1 Setup

Consider a small open economy inhabited by a continuum of identical consumers located in a unit interval. A consumer's lifetime expected utility is $ \mathbb{E}_{t}\sum_{j=0}^{\infty}\beta^{j}u\left(C_{t+j}\right) $ where $ C_{t} $ denotes consumption and $ \beta $ is the discount factor satisfying $ \beta\in(0,1) $ . Also, $ E_{t}\left[\cdot\right] $ represents the
expectations operator conditional on the information available in period t. Following Uribe and Schmitt-Grohé (2017), we specify $ u(\cdot) $ as $ u(C_{t}) = -0.5 \left( C_{t} - \bar{C} \right)^{2} $ , where $ \bar{C} $ is the bliss point so that $ C_{t} < \bar{C} $ for all t. The consumers' flow budget constraint is

$$ A_{t+j+1}=(1+r)A_{t+j}+X_{t+j}-C_{t+j}, $$

for all j > 0. Here, $ A_{t} $ denotes foreign asset holdings at the beginning of period t. In this budget constraint, consumers have access to perfect international capital markets where capital can be saved or borrowed at a constant world interest rate r. To facilitate the analysis, we assume that the rate of time preference is equal to $ r: \beta = 1/(1 + r) $ . Finally, $ X_{t} $ is the exogenous stochastic endowment which we can also interpret as net output.

The first-order condition implies that $ C_{t} = E_{t}C_{t+j} $ for $ j = 1, 2, \ldots $ . The optimal consumption is given by

$$ C_{t}=r A_{t}+X_{t}^{p}, $$

where $ X_{t}^{p} $ is the non-financial permanent income given by $ X_{t}^{p} = [r / (1 + r)] \sum_{j=0}^{\infty} (1 + r)^{-j} \mathbb{E}_{t} X_{t+j} $ . Changes in consumption are calculated as $ \Delta C_{t+1} = \Delta E_{t+1} X_{t+1}^{p} $ , where $ \Delta E_{t+1} $ denotes a change in the expectations operator defined by $ \Delta E_{t+1} = E_{t+1} - E_{t} $ . This equation means that consumption between periods t and $ t+1 $ changes only if changes in the permanent income are recognized by consumers.

By the fundamental balance-of-payments identity, the current account $ CA_{t} $ equals changes in the country's net foreign assets (i.e., $ CA_{t} \equiv A_{t+1} - A_{t} $ ). Therefore, using (1) and (2), we have

$$ C A_{t}=X_{t}-X_{t}^{p}=-\sum_{j=1}^{\infty}\left(1+r\right)^{-j}\mathbb{E}_{t}\Delta X_{t+j}, $$

meaning that the current account is minus the present discounted value of future expected changes in the net output.

It is straightforward to extend this standard RE model with imperfect capital mobility. Shibata and Shintani (1998) consider consumers with limited access to the international capital market, as in Campbell and Mankiw (1989, 1990). A fraction $ 1 - \lambda $ of consumers have access to international capital markets and their consumption is determined by (2). The remaining fraction $ \lambda $ of consumers hold neither international financial assets nor liabilities (i.e., financial autarky): They are hand-to-mouth consumers who consume all endowment within a single period. We call $ \lambda $ the degree of imperfect capital mobility and this model the hybrid RE model. Denoting the aggregate consumption in the hybrid model by $ C_{t}^{HY} $ , we express $ C_{t}^{HY} = (1 - \lambda) C_{t} + \lambda X_{t} $ . Likewise, let $ CA_{t}^{HY} $ be the current account in the hybrid
model. Because hand-to-mouth consumers do not hold international financial assets, $ CA_{t}^{HY} $ becomes

$$ CA_{t}^{HY}=\left(1-\lambda\right)CA_{t}. $$

### 3.2 Characterizing the RE models

We characterize the pure and hybrid RE models using (3) and (4). We look at theoretical predictions for the persistence and volatility of the current account under the assumption of AR(1) process of the net output growth.

Let $ g_{t} \equiv \ln \left( X_{t}/X_{t-1} \right) $ be the net output growth. Assume that $ g_{t} = (1 - \phi) \mu + \phi g_{t-1} + \varepsilon_{t} $ , where $ \varepsilon_{t} \sim i.i.d. \left( 0, \sigma^{2} \right) $ and $ |\phi| < 1 $ . As discussed in Campbell and Deaton (1989), this assumption of the stationary net output growth rate requires some reformulation of variables. In particular, to ensure the stationarity of the current account, we divide both sides of (3) by $ X_{t} $ . We obtain $ ca_{t} \equiv CA_{t}/X_{t} = -\sum_{j=1}^{\infty} (1 + r)^{-j} E_{t} \left( \Delta X_{t+j}/X_{t} \right) $ . While we will evaluate the model with $ \mu \neq 0 $ in Section 5, we temporarily set $ \mu = 0 $ only for expositional purposes. In this case, the first-order approximation of $ ca_{t} $ yields

$$ c a_{t}\simeq-\sum_{j=1}^{\infty}\left(1+r\right)^{-j}\mathbb{E}_{t}g_{t+j}=-\frac{\phi}{1+r-\phi}g_{t}. $$

The left column of Table 2 summarizes the characterization of the RE model's predictions. The proportionality of $ ca_{t} $ to $ g_{t} $ in (5) implies that the persistence of $ ca_{t} $ should be the same as that of $ g_{t} $ , namely, $ \rho_{ca} = \phi $ . It also implies that the predicted volatility is given by $ V_{ca} = sd(ca_{t}) / sd(g_{t}) = \phi / (1 + r - \phi) $ . From Hall's random walk hypothesis, it is evident that changes in consumption are serially uncorrelated in the RE model. In our case with reformulation of variables, we define $ \Delta c_{t+1} \equiv \Delta C_{t+1} / X_{t} $ . These newly defined (normalized) changes in consumption can be expressed as $ \Delta c_{t+1} = \Delta E_{t+1} \left( X_{t+1}^{p} / X_{t} \right) \simeq \left[ (1 + r) / (1 + r - \phi) \right] \varepsilon_{t+1} $ by the first-order approximation. Since $ \varepsilon_{t} $ is independent and identically distributed, the predicted persistence of changes in consumption is zero, namely, $ \rho_{\Delta c} = 0 $ .

These predictions of the pure RE model are inconsistent with the evidence in Section 2. While the model predicts that $ \rho_{ca} = \phi $ , the data suggests that $ \rho_{ca}^{data} > \phi^{data} $ , as indicated in Table 1. It also fails to explain the volatility of the current account, if we recall $ V_{ca}^{data} \simeq 2 $ . For the pure RE model to generate $ V_{ca} \simeq 2 $ , taking the world interest rate r as given, it requires a sufficiently large value of $ \phi $ . For example, if r = 0.04, $ \phi $ needs to be as large as 0.69, as opposed to the data of $ \phi^{data} = 0.16 $ . $ ^{13} $ Note also that serially uncorrelated changes
in consumption, $ \rho_{\Delta c}=0 $ , is against the data, $ \rho_{\Delta c}^{data}>0 $ .

The hybrid RE model can generate better predictions for the persistence of changes in consumption (see the right column of Table 2). Again, denoting $ \Delta c_{t+1}^{HY} \equiv \Delta C_{t+1}^{HY} / X_t $ , we have $ \Delta c_{t+1}^{HY} \simeq (1 - \lambda) \Delta c_{t+1} + \lambda g_{t+1} $ in the hybrid RE model. Since $ g_{t+1} $ is serially correlated, the assumption of imperfect capital mobility leads to predictable changes in consumption. The resulting $ \rho_{\Delta c}^{HY} $ takes a form of $ \rho_{\Delta c}^{HY} = \Lambda_0 \phi \neq 0 $ , where $ \Lambda_0 $ is a function of $ \lambda $ and the second moments of $ \Delta c_t $ and $ g_t $ (see the footnote of Table 2 for the analytical expression). The improvement can be made at least without influencing the predicted persistence of the current account. Because $ CA_t^{HY} = (1 - \lambda) CA_t $ from (4), $ ca_t^{HY} = (1 - \lambda) ca_t = -(1 - \lambda) \phi / (1 + r - \phi) g_t $ , where $ ca_t^{HY} \equiv CA_t^{HY} / X_t $ . The proportionality of $ ca_t^{HY} $ to $ ca_t $ means that the first-order autocorrelation of $ ca_t^{HY} $ is unchanged in comparison to the pure RE model: $ \rho_{ca}^{HY} = \rho_{ca} = \phi $ .

However, the better prediction for the persistence of changes in consumption is achieved at the cost of undermining the predicted volatility of the current account. The above proportionality also implies that $ V_{ca}^{HY} = (1 - \lambda) V_{ca} $ , where $ V_{ca}^{HY} \equiv sd \left(ca_{t}^{HY}\right) / sd \left(g_{t}\right) $ . Since $ \lambda \in [0, 1) $ , the volatility of $ ca_{t}^{HY} $ is dampened by $ \lambda $ . Therefore, for the hybrid RE model to generate $ V_{ca} \simeq 2 $ , the hybrid model needs an even larger value of $ \phi $ than the pure RE model. For example, if r = 0.04 and $ \lambda = 0.50 $ , the hybrid RE model requires that $ \phi = 0.83 $ , which is larger than 0.69 required in the pure RE model. $ ^{14} $

## 4 The SI models

To improve predictions of the ICA model, we replace RE in the ICA model with SI. In this section, we leave the detailed maximization problem of the SI model to Appendix A.1 and focus on how the current account dynamics are described in the SI model.

### 4.1 Setup

Let us assume that, in every period, only randomly selected $ 1 - \omega $ fraction of consumers update their information set and that the remaining fraction $ \omega $ of consumers do not update their information set. We call $ \omega $ the degree of information rigidity. Our assumption simplifies the SI model of consumption by Reis (2006), who considers endogenous infrequent informa-
tion updating. $ ^{15} $ Despite the simplification, as we will later show, the model predictions are very similar to Reis (2006).

Given infrequently updated information, consumers make decisions as rationally as they can. Suppose that a consumer updated his information in period t and does not obtain new information in $ t + 1 $ . In this case, he does not change his consumption in period $ t + 1 $ since he recognizes no changes in the permanent income. He sticks to a consumption plan that he could make in period t. Therefore, if the information is not updated in period $ t + 1 $ , any shock in period $ t + 1 $ is unrecognized and absorbed by the consumer's saving. Reis (2006) refers to consumers who stick to their consumption plans as inattentive consumers.

More formally, an inattentive consumer who has period-t information chooses the ex ante optimal plan of consumption $ \{C_{t+j,j}\}_{j=0}^{\infty} $ , where $ C_{t+j,j} $ is consumption chosen for period $ t+j $ with j-period delay of information. Note that $ C_{t,0} $ represents consumption that the consumer enjoys in period t with full information. This $ C_{t,0} $ must be the same as the optimal consumption under the RE model. If the consumer does not receive new information until period $ t+j $ , the consumer chooses $ C_{t+j,j} $ that was planned in period t. Note also that $ C_{t+j,j}=C_{t,0} $ is satisfied because the inattentive consumer perceives no changes in his permanent income. Therefore, for $ j=1,2,\ldots $ , we have

$$ C_{t+j,j}=C_{t,0}=r A_{t}+X_{t}^{p}. $$

Let $ S_{t+j,j} $ be changes in the foreign asset holdings of the consumer in period $ t+j $ , conditional that his information was updated in period t but has not been updated until period $ t+j $ . Then, $ S_{t+j,j} $ is given by

$$ S_{t+j,j}=r A_{t+j}+X_{t+j}-C_{t+j,j}, $$

where $ A_{t+j} $ follows (1) evaluated at $ C_{t+j}=C_{t+j,j} $ . That is, $ A_{t+j+1}=(1+r)A_{t+j}+X_{t+j}-C_{t+j,j} $ .

While $ S_{t+j,j} $ in (7) looks similar to the current account in the RE model, they need to be distinguished in two aspects. First, $ S_{t+j,j} $ absorbs all changes in endowment between t and $ t+j $ , together with returns of the unintended change in the foreign asset holdings. Using (6), we can rewrite $ S_{t+j,j} $ as

$$ \begin{align*}S_{t+j,j}\ =&r\left(A_{t+j}-A_{t}\right)+X_{t+j}-X_{t}^{p}\\=&r\left(\Delta A_{t+j}+\ldots+\Delta A_{t+1}\right)+\Delta X_{t+j}+\ldots+\Delta X_{t+1}+S_{t,0},\mathrm{for}j=0,1,2,\ldots,\end{align*} $$
where $ S_{t,0} = X_t - X_t^p $ which equals the current account under the RE model. Under SI, the inattentive consumer’s consumption $ (C_{t+j,j}) $ does not respond to the unrecognized changes in endowment between periods t and $ t+j $ . In (8), saving absorbs all unrecognized changes in endowment $ \Delta X_{t+1}, \Delta X_{t+2}, \ldots, \Delta X_{t+j} $ . Accordingly, changes in the inattentive consumer’s foreign asset holdings also include returns from the unrecognized changes in endowment (i.e., $ r(\Delta A_{t+j} + \ldots + \Delta A_{t+1}) $ ). We rewrite (8) recursively as $ ^{16} $

$$ S_{t+j,j}=\left(1+r\right)S_{t+j-1,j-1}+\Delta X_{t+j}for j=1,2,3,\ldots $$

The second aspect that should be distinguished from the current account in the RE model is that changes in foreign asset holdings differ across inattentive consumers, depending on how they update their information. To obtain the current account in the SI model, we need to aggregate individual foreign asset holdings across all inattentive consumers. Based on the assumption of information updating, the current account in the SI model is

$$ C A_{t}=\left(1-\omega\right)\sum_{k=0}^{\infty}\omega^{k}S_{t,k}, $$

where $ S_{t,k} $ is the period-t changes in the inattentive consumer’s foreign assets based on the information in period t-k. Using (9), $ S_{t,k} $ can also be written as

$$ S_{t,k}=\left(1+r\right)S_{t-1,k-1}+\Delta X_{t}, $$

where $ S_{t-k,0} = X_{t-k} - X_{t-k}^{p} $

It is again straightforward to extend the SI model with imperfect capital mobility. The aggregate consumption is given by $ C_{t}^{HY} = (1 - \lambda) C_{t} + \lambda X_{t} $ and the current account is given by

$$ CA_{t}^{HY}=\left(1-\lambda\right)CA_{t} $$

in the hybrid SI model so that (4) continues to hold. $ ^{17} $
### 4.2 Characterizing the SI models

We characterize the SI and the hybrid SI models using $ (10)-(12) $ . We establish propositions for the current account and changes in consumption under the pure SI model. We then discuss the hybrid SI model.

#### 4.2.1 Persistence and volatility of the current account

The following proposition describes current account dynamics under the pure SI model:

Proposition 1 Suppose that inattentive consumers update their information with the probability of $ 1 - \omega $ every period. Suppose also that the degree of information rigidity $ \omega $ is sufficiently low such that $ \omega(1 + r) < 1 $ . Then, the current account is given by

$$ C A_{t}=\omega^{*}C A_{t-1}+\left(1-\omega\right)S_{t,0}+\omega\Delta X_{t}, $$

where $ S_{t,0}=X_{t}-X_{t}^{p} $ and $ \omega^{*}=\omega(1+r) $ .

Furthermore, suppose that the net output growth follows a covariance-stationary $ AR(1) $ process with mean zero: $ g_{t} = \phi g_{t-1} + \varepsilon_{t} $ , where $ \varepsilon_{t} \sim i.i.d. $ $ (0, \sigma^{2}) $ . Then, $ c_{a_{t}} $ can be approximated by the $ AR(2) $ process:

$$ c a_{t}=\left(\omega^{*}+\phi\right)c a_{t-1}-\left(\omega^{*}\phi\right)c a_{t-2}+\frac{\omega^{*}-\phi}{1+r-\phi}\varepsilon_{t}. $$

Proof: See the Appendix A.3. □

The first part of Proposition 1 tells us that the current account is written as the first-order difference equation with the two driving forces. The first driving force is $ S_{t,0} $ , changes in foreign asset holdings under full information. The impact of $ S_{t,0} $ becomes weaker as $ \omega $ becomes higher. The second driving force is $ \Delta X_{t} $ . A fraction $ \omega $ of inattentive consumers with old information let their saving absorb unrecognized shocks to endowment. For this reason, $ \Delta X_{t} $ appears in (13) with its coefficient $ \omega $ . Also, recall that the effect of unrecognized changes in endowment on $ CA_{t} $ is carried over to the subsequent periods with interest earnings. Therefore, as $ \omega^{*} = \omega(1 + r) $ increases, the coefficient on $ CA_{t-1} $ becomes larger, making $ CA_{t} $ more persistent. Turning to the second part of the proposition, $ ca_{t} $ follows the AR(2) process, generalizing (5) in the RE model. Indeed, substituting $ \omega^{*} = \omega(1 + r) = 0 $ in (13) results in the AR(1) process $ ca_{t} = \phi ca_{t-1} - \phi / (1 + r - \phi) \varepsilon_{t} $ . Comparing $ ca_{t} = -\phi / (1 + r - \phi) \sum_{j=0}^{\infty} \phi^{j} \varepsilon_{t-j} $ with $ g_{t} = \sum_{j=0}^{\infty} \phi^{j} \varepsilon_{t-j} $ implies $ ca_{t} = -\phi / (1 + r - \phi) g_{t} $ , reproducing (5).
The persistence of $ ca_{t} $ is now increasing in $ \omega^{*} $ so that the pure SI model can generate a persistent current account even when net output growth is not persistent. In particular, the first-order autocorrelation of $ ca_{t} $ is calculated as $ (\omega^{*}+\phi)/(1+\omega^{*}\phi) $ from (14) as long as $ \omega^{*}\neq\phi $ . The left panel of Figure 1 shows the first-order autocorrelation of the current account against $ \omega^{18} $ . The blue line is the persistence under $ \phi=0.20 $ , while the red line is that under $ \phi=0.80 $ . We reconfirm that persistence of the current account increases with $ \omega $ , regardless of the value of $ \phi $ , except for the case of $ \omega^{*}=\phi $ . If $ \omega^{*} $ happens to equal $ \phi $ , the coefficient on $ \varepsilon_{t} $ is zero in (14) and the current account is constant for all t. Our result on the persistence of the current account can be summarized in the next corollary:

Corollary 1 Under the pure SI model with $ \mu = 0 $ , the first-order autocorrelation of the current account $ c_{t} $ is given by

$$ \rho_{c a}=\frac{\omega^{*}+\phi}{1+\omega^{*}\phi}, $$

and is always greater than $ \rho_{ca} = \phi $ under the pure RE model, as long as $ \omega^{*} \neq \phi $ .

We move on to volatility of the current account in the SI model. Equation (14) implies the following corollary:

Corollary 2 Under the pure SI model with $ \mu = 0 $ , the volatility of the current account $ V_{ca} = sd(ca_t) / sd(g_t) $ is given by

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

The middle panel of Figure 1 plots the volatility against $ \omega $ . The volatility decreases with $ \omega $ if $ \omega^{*} < \phi $ and increases with $ \omega $ if $ \omega^{*} > \phi $ . When $ \phi = 0.20 $ , the volatility increases with $ \omega $ over a wide range of $ \omega $ . Therefore, even when $ \phi $ takes a low value as in the data, the SI model can generate a volatile current account with a large value of $ \omega $ . By contrast, as shown in the red line, when $ \phi = 0.80 $ , the volatility increases with $ \omega $ over a very narrow range of $ \omega $ .

#### 4.2.2 Persistence of consumption growth

It is worthwhile discussing persistent changes in consumption in the SI model. Using the model with endogenous infrequent information updating, Reis (2006) demonstrates that his
SI model of consumption can successfully generate predictable changes in consumption. We confirm persistent changes in consumption under exogenous infrequent information updating. In the next proposition, we analytically derive the stochastic process of $ \Delta c_{t+1} $ :

Proposition 2 Suppose that inattentive consumers update their information with the probability of $ 1 - \omega $ every period. Suppose also that the degree of information rigidity $ \omega $ is sufficiently low such that $ \omega(1 + r) < 1 $ . Then, changes in the consumption are given by

$$ \Delta C_{t+1}=\omega^{*}\Delta C_{t}+\left(1-\omega\right)\Delta\mathbb{E}_{t+1}X_{t+1}^{p}. $$

Furthermore, suppose that net output growth follows a covariance-stationary $ AR(1) $ process with mean zero: $ g_{t} = \phi g_{t-1} + \varepsilon_{t} $ , where $ \varepsilon_{t} \sim i.i.d. $ ( $ 0, \sigma^{2} $ ). Then, $ \Delta c_{t+1} $ can be approximated by the $ AR(1) $ process:

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

Proof: See the Appendix A.4. □

As is clear in (18), the persistence of changes in consumption critically depends on $ \omega^{*} $ . In the right panel of Figure 1, we plot the first-order autocorrelation of $ \Delta c_{t} $ . It has a one-to-one relationship with $ \omega $ . For example, when $ \omega = 0.80 $ and r = 0.04, $ \rho_{\Delta c} = (1 + r)\omega = 0.83 $ . We summarize this result in the following corollary:

Corollary 3 Under the pure SI model with $ \mu=0 $ , the first-order autocorrelation of $ \Delta c_{t} $ is given by $ \rho_{\Delta c}=\omega^{*} $ , and is always greater than $ \rho_{\Delta c}=0 $ under the pure RE model.

We have two additional remarks on changes in consumption. First, not surprisingly, the dynamic properties of changes in consumption are very close to those in Reis (2006). His proposition 2 states that, when the maximum length of time during which consumers are inattentive is q periods, changes in consumption follow the MA(q) process with monotonically decreasing MA coefficients. In our SI model, the maximum length of time of being inattentive is infinity and the MA( $ \infty $ ) representation of the AR(1) process (18) implies that MA coefficients are exponentially decaying.

Second, as Reis (2006) argues, the SI model can provide a microfoundation for the representative-agent habit formation model in which persistent changes in consumption are well explained. It can analytically be shown that the current account in the SI model is the observational equivalent to the current account in the habit formation model. In particular,