Open Economy Macroeconomics: Theory, methods and applications ...

Open Economy Macroeconomics: Theory,

methods and applications

Econ PhD, UC3M

Lecture 9: Data and facts

Hernán D. Seoane

UC3M

Spring, 2016

Today’s lectureA look at the data

• Study what data says about open economies

• Stress 1: business cycle differences among developed andemerging economies

• Stress 2: private versus sovereign behavior

• Technical point: data treatment (log-linear, log-quadratic, HPfilter, Band pass filter, growth rates, etc)

Today’s lecture (cont.)A Mickey Mouse model

• Endowment small open economy

• Small open economy with capital

• RBC small open economy model

Today’s lectureReferences

• This lecture is based on chapters 1 to 4 of Schmitt-Grohe andUribe (2015) texbook

Detrending

• We focus in cyclical component

• We have to remove the trend or secular component from the rawdata

• The cyclical component is going to be different depending on theassumptions about the trend

• yt = yct + ys

t

DetrendingLog-quadratic trend

• yt denotes the log of real output (per capita)

• Assume: yt = a + bt + ct2 + εt

• Set yct = εt

• yst = a + bt + ct2


Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1




• The cyclical component shows 3 cycles

• std(yct) = 10.7%

• corr(yct , yc

t−1) = 0.85


• 94 countries,at least 30 observations, worldwide

• Cross country average std(yt) = 6.2%

• std(gt)/std(yt) = 2.3

• Government consumption is more than 2 times more volatilethan output





• ρ(y, tby) = −0.15 and ρ(y, g) = −0.02

Facts

1 High global volatility

2 High volatility of government consumption

3 Global rank of volatilities

4 Procyclical aggregate demand components

5 Countercyclical trade balance and the current account

6 Acyclical Government consumption to GDP

7 Persistence



Facts

1 US is much less volatile than the rest of the world

2 std(ct)/std(yt) = 1.02... the role of durable goods

3 std(gt)/std(yt) = −0.32... government spending is stronglycounter-cyclical

4 US is much less open than the rest of the world,(x + m)/y = 18%, while for the world is about twice as much

5 Conditional on income level





1 Conditional on country size

Facts Emerging Economies

1 Excess volatility of emerging and poor countries

2 Less consumption smoothing

3 Countercyclical trade balance increases with income

4 Countercyclical government spending increases with income

DetrendingHodrick-Prescott Filter (1987)

• Here, the trend and cyclical components are identified by solvinga minimization problem

min{yct ,ys

t}Tt=1

{T

∑t=1

(yct)

2 + λT−1

∑t=2

[(ys

t+1 − yst)−(ys

t − yst−1)]2}

• Subject to yt = yct + ys

t


• Trade-off between minimizing the variance of the cyclicalcomponent and keeping the growth rate of the trend constant

• λ regulates this trade-off. The higher λ, the more we penalizechanges in the growth rate of the trend component. If λ isinfinite the resulting trend is linear

• Ravn and Uhlig (2001)





Three Models

1 Cycles look very different

2 However, business cycle facts still hold

3 Other options: first differencing

4 However, business cycle facts still hold

5 Quarterly data too

Three Models

1 Small Open Economy Endowment model

2 Small Open Economy with Capital

3 Small Open Economy Real Business Cycle Model

SOE Endowment model

E0

∞

∑t=0

βtu(ct)

dt − dt−1 = rdt−1 + ct − yt

• dt is the debt position (net) assumed in period t and due in t + 1

• A no Ponzi condition limj→∞

Ejdt+j

(1+r)j ≤ 0

• The Lagrangian for this problem is

L = E0

∞

∑t=0

βt {u(ct) + λt [dt − dt−1 − rdt−1 − ct + yt]}

SOE Endowment model

• Euler equation is

u′(ct) = β(1 + r)Et[u′(ct+1)]

• with the standard interpretations: in the margin the household isindifferent between consuming in period t or saving the extra unit ofconsumption for period t + 1

• The intertemporal budget constraint

(1 + r)dt−1 = Et

∞

∑j=0

yt+j − ct+j

(1 + r)j

SOE Endowment model

• Note that the definition of the trade balance is

tbt = yt − ct

• The intertemporal budget constraint implies then that

(1 + r)dt−1 = Et

∞

∑j=0

tbt+j

(1 + r)j

SOE Endowment model

• Assume u(ct) = − 12 [ct − c]2 and β(1 + r) = 1

• Euler equation isct = Et[ct+1]

• Standard RW result

• Using the intertemporal budget constraint we can get

ct =r

(1 + r)Et

∞

∑j=0

yt+j

(1 + r)j − rdt−1

• note that it depends on the the income process

SOE Endowment model

• Note, r(1+r)Et ∑∞

j=01

(1+r)j = 1

• Consumption is a weighted average of your expected lifetimestream of endowments

• Endowments are exogenous

• dt−1 is predetermined

ct =r

(1 + r)Et

∞

∑j=0

yt+j

(1 + r)j − rdt−1

• is a closed form solution of ct

• Define ypt = r

(1+r)Et ∑∞j=0

yt+j

(1+r)j

SOE Endowment model• Plugging the closed form solution to ct in the budget constraint,

we getdt − dt−1 = yp

t − yt

• is a closed form solution of debt

• An expression for the current account

cat = tbt − rdt−1

• Combining it with the sequential budget constraint

cat = −(dt − dt−1)

• The current account equals the change in the net foreign assetposition (a deficit in the current account is associated with anincrease in foreign debt)

SOE Endowment model

• we can also writecat = yt − yp

t

• you run a current account surplus when income is larger thanyour permanent income

• Etyt+j = ρjyt

• Alsotbt = yt − yp

t − rdt−1

SOE Endowment model

• Recall that we assume endowment follows a mean revertingAR(1) process

• Etyt+j = ρjyt

• Herect =

r(1 + r− ρ)

yt − rdt−1

• Consumption responds positively to income, but not 1 to 1

• Recall: tbt = yt − ct

• Recall: cat = rdt−1 + tbt or cat = −(dt − dt−1)

SOE Endowment model

• you can build the path of sovereign debt

dt = dt−1 −1− ρ

(1 + r− ρ)yt

• There is a unit root for the international assets

• The trade balance

tbt = rdt−1 +1− ρ

(1 + r− ρ)yt

cat =1− ρ

(1 + r− ρ)yt

• A positive shock to output have a permanent effect on foreigndebt and a permanent deterioration on the trade balance

SOE Endowment model: stationary

shock

Endowment

Trade Balance and the Current Account

Foreign Debt

Consumption

ca1

tb1

SOE Endowment model

• If ρ→ 1, you tend to consume all the increase in endowment

• No changes in your debt position, trade balance or currentaccount

SOE Endowment model

• Problem, current account is pro-cyclical (it improves duringexpansions)

• Trade balance is procyclical

• In the data it is the opposite

• What happens with permament shocks?

• ∆yt = yt − yt−1 and ∆yt = ρ∆yt−1 + εt

SOE Endowment model

• Implies

cat = −Et

∞

∑j=1

∆yt+j

(1 + r)j

• ca equals expected output fall. In we expect output to fall, ca ispositive because we are saving (accumulating assets) for futureconsumption smoothing

• Given that Et∆yt+j = ρj∆yt

cat =−ρ

1 + r− ρ∆yt

SOE Endowment model:

Nonstationary shock

Consumption

Output

Source: Uribe (2014).

SOE Endowment model

• Consumption responds on impact more than output

• The difference is financed with foreign debt

• Implies that current account deteriorates

• Trade balance also deteriorates

• This is more in line with the data

SOE Endowment modelTesting the model

• A testable implication of the model

cat = −∞

∑j=1

Et∆yt+j

(1 + r)j

• Let x =

[∆yt

cat

]• Estimate the following VAR

xt = Dxt−1 + εt

• Under ∆yt univariate AR(1), the VAR representation exists


• Use Ht to denote the information contained in vector xt, thenEt[xt+j|Ht] = Djxt

• Using this representation

∞

∑j=1

Et[∆yt+j|Ht]

(1 + r)j = [1 0][

I− D1 + r

]−1 D1 + r

[∆yt

cat

]

• Use

F = −[1 0][

I− D1 + r

]−1 D1 + r


• Now, suppose you run 2 separate regressions on the right andleft hand sides of

cat = −∞

∑j=1

Et∆yt+j

(1 + r)j

• The coefficients for the regressors for the left hand side should be[0 1]

• The coefficients for the regressors for the right hand side shouldbe F

• The model implies the constraint than F = [0 1]

• Nason and Rogers (2006) test this implications using Canadiandata

SOE Endowment model

1 The data rejects this cross-equation restriction

2 This suggests we should start looking for other model

3 How do things change when we endogenize output

4 Introduce capital... why capital?

5 If investment is procyclical, this will contribute to acountercyclical trade balance

Capital and open economy

E0

∞

∑t=0

βtu(ct)

ct + it + (1 + r)dt−1 = yt + dt

yt = θtF(kt)

kt+1 = kt + it

limj→∞dt+j

(1 + r)j ≤ 0

• And the no Ponzi game constraint


• Optimality conditions

u′(ct) = λt

λt = β(1 + r)λt+1

λt = βλt+1[1 + θt+1F′(kt+1)]

ct + kt+1 − kt + (1 + r)dt−1 = dt + θtF(kt)


• Set β(1 + r) = 1

ct = ct+1

r = θt+1F′(kt+1)

ct + rdt−1 =r

(1 + r)

∞

∑j=0

θt+jF(kt+j)− kt+j+1 + kt+j

(1 + r)j

• Implies

kt+1 = κ

(θt+1

r

); κ′ > 0

Capital and open economySteady state equilibrium

• Let’s look at the steady state equilibrium

• Until t = −1 the technology θt is fixed at θ

• Agent’s expect θt to remain at this level

• However, in this period θt = θ′ > θ for all t ≤ 0

• Recall that k was constant at k, now

k = κ

(θ

r

)< κ

(θ′

r

)= k′

• So, investment jumps only for one period


• Output increases because of the increase in productivity

• Later, it increases once more because of the increase in the capitalstock

• We can plug these dynamics in the budget constraint



Capital and open economyA permanent shock

• Note that k0 = k as it was chosen in t = −1, but k1 increases

• This implies that i0 = k1 − k0 = k′ − k. Plug it here

c0 = −rd +r

(1 + r)[θ′F(k)− k′ + k] +

1(1 + r)

θ′F(k′)

• or, given that all future consumption is constant

c′ = −rd + θ′F(k) +1

1 + r{

θ′[F(k′)− F(k)]− r(k′ − k)}


• or

c′ = −rd + θ′F(k) +1

1 + r{

θ′[F(k′)− F(k)]− θ′F′(k′)(k′ − k)]}

• Given that F is strictly concave and k′ > k, the bracket is positive( F(k′)−F(k)

k′−k > F′(k′))

• Thenc′ > −rd + θ′F(k) > −rd + θF(k) = c


• It can be shown that consumption increases more than output inperiod 0

• Intuition, output will continue to grow after 0 because of theresponse of capital

• Households increase borrowing

• Trade balance deteriorates to finance the extra investment andthe extra consumption

• The current account also deteriorates and after period 1 it goesback to 0

• this result is different from the endowment economy, with apermanent shock trade balance used to remain unchanged, buthere deteriorates on impact

Capital and open economyA temporary shock

• Suppose that now, unexpectedly θ increases but only for 1period; i.e. purely temporal change

• Investment does not respond. We will only have high technologyfor 1 period, but capital is fixed in that period

• Output then only changes because of the productivity change,and only for that first period

y0 = y−1 + (θ′ − θ)F(k)

• Here y−1 = AF(k). Remember that consumption was equal to(before the shock)

c−1 = −rd + θF(k)

Capital and open economyA temporary shock

• Now you can show that

c0 = c−1 +r

1 + r(y0 − y−1)

• Implications of the permanent income hypothesis are thatconsumption responds only partially (you only consume thereturns of extra income, and reinvest the rest)

• savings increases, so trade balance has to improve

tb0 − tb−1 = (y0 − y−1)− (c0 − c−1) =1

1 + r(A′ − A)F(k) > 0

• persistence of the shock affects the countercyclicality of the tradebalance

Capital and open economyAdjustment costs

• usually the open economy model needs to assume adjustmentcosts on capital, otherwise the growth rate of investment iscounterfactually large during the period of the shock

(1 + r)dt−1 = dt + θtF(kt)− ct − it −12

i2tkt

• Investment does not respond

• Output then only changes because of the productivity change, and onlyfor that first period

• Implications of the permanent income hypothesis are that consumptionresponds only partially

• savings increases, so trade balance has to improve


E0

∞

∑t=0

βtu(ct)

(1 + r)dt−1 = dt + yt − ct − it −12

i2tkt

yt = θtF(kt)

kt+1 = kt + it

limj→∞dt+j

(1 + r)j ≤ 0

• And the no Ponzi game constraint


• Optimality conditions. Denote βtλt the Lagrange multiplier onthe budget constraint and βtλtqt the Lagrange multiplier on thelaw of motion of capital

L =∞

∑t=0

βt

{u(ct) + λt

[θtF(kt) + dt − (1 + r)dt−1 − ct − it −

12

i2tkt

+ qt(kt + it − kt+1)

]}

• which gives the following FOCs

1 +itkt

= qt

λtqt = βλt+1

[qt+1 + θt+1F′(kt+1) +

12

(it+1kt+1

)2]


• Keep assuming that β(1 + r) = 1

• This implies that ct = ct+1 and λt is constant

ct = ct+1

ct = rbt−1 +r

(1 + r)

∞

∑j=0

θt+jF(kt+j)− it+j − 12

i2t+jkt+j

(1 + r)j


1 +itkt

= qt

(1 + r)qt = θt+1F′(kt+1) +12

(it+1kt+1

)2+ qt+1

it = kt+1 − kt

• Here qt is the Tobin’s q: 1 + itkt= qt implies that the marginal cost of

producing 1 unit of capital equals the marginal revenue of selling it

• the following equation is an arbitrage condition that compares thereturn of that unit of capital (sold) in bonds, and the returns on keepingthe unit of capital and using it for production


• Working out the FOC, you can obtain a dynamic system forcapital and q

kt+1 = qtkt

qt =θt+1F′(qtkt) + (qt+1 − 1)2/2 + qt+1

1 + r




• A permanent shock

• The locus KK’ does not change

• QQ’ shifts up

• capital increases permanently but the price does not change inthe long run

• now capital adjusts slowly, before without adjustment costs thecapital stock goes to the steady state level immediately

• Then the trade balance is less responsive to productivity shocks


1 Proposition 1: The more persistence the productivity shock, themore likely is the trade balance to deteriorate after a productivityshocks

2 Proposition 2: the higher adjustment costs of capital, the lesslikely the trade balance is going to deteriorate after aproductivity shock

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