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
DetrendingLog-quadratic trend
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
DetrendingLog-quadratic trend
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
DetrendingLog-quadratic trend
• The cyclical component shows 3 cycles
• std(yct) = 10.7%
• corr(yct , yc
t−1) = 0.85
DetrendingLog-quadratic trend
• 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
DetrendingLog-quadratic trend
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
DetrendingLog-quadratic trend
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
• 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
DetrendingLog-quadratic trend
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
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
DetrendingLog-quadratic trend
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
DetrendingLog-quadratic trend
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
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
DetrendingHodrick-Prescott Filter (1987)
• 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)
DetrendingHodrick-Prescott Filter (1987)
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
DetrendingHodrick-Prescott Filter (1987)
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch1
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
SOE Endowment modelTesting the model
• 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
SOE Endowment modelTesting the model
• 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
Capital and open economy
• 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)
Capital and open economy
• 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
Capital and open economySteady state equilibrium
• 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 economySteady state equilibrium
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch3
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)}
Capital and open economyA permanent shock
• 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
Capital and open economyA permanent shock
• 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
Capital and open economy
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
Capital and open economy
• 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]
Capital and open economy
• 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
Capital and open economy
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
Capital and open economy
• 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
Capital and open economy
Figure: Source: Schmitt-Grohe and Uribe (2015) Ch3
Capital and open economy
• 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
Capital and open economy
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