Study of asset returns in SGM

Naik, V. 1994. "Asset Prices in Dynamic Production Economies with Time-Varying Risk." Review of financial studies, 7(4), 781-801.

Model
(i)   time-varying uncertainty of the marginal product of capital (productivity shocks),
(ii)  the presence of capital adjustment costs,
(iii) the separation of intertemporal substitution and risk aversion (EZ preferences), and
(iv) the allowance for first-order risk aversion in investors' preferences.

Results
(i)   the sensitivity of the price of the aggregate capital stock to shifts in risk is crucially dependent on the level of capital adjustment costs.  The effect of shifts in uncertainty is significant only when capital adjustment costs are substantial.
(ii)  Intertemporal substitution governs the direction of the effect of changes in risk on the value ofcapital stock.  Risk aversion is important in the determination of the magnitude of this effect.


Campanale, C.; R. Castro and G. L. Clementi. 2010. "Asset Pricing in a Production Economy with Chew-Dekel Preferences." Review of Economic Dynamics, 13(2), 379-402.

<quote>
Tallarini (2000) showed that the first of the two issues just outlined can be addressed by disentangling risk aversionfrom the elasticity of intertemporal substitution. By assuming Epstein–Zin preferences, he was able to raise risk aversion at arbitrarily high levels, while keeping the elasticity of substitution anchored at 1. Tallarini went on to show the existence of RRA coefficients such that the market price of risk is consistent with the empirical evidence. However, the price of capital being constant at 1, his model essentially generates no equity premium. This issue was dealt with successfully by Jermann (1998) and Boldrin et al. (2001), who assumed that the allocation of capital cannot adjust immediately or costlessly to productivity shocks. Impediments to the smooth adjustment of capital imply that its price may vary away from 1. However, without limiting households’ willingness to substitute consumption intertemporally, this innovation results mainly in a higher volatility of consumption growth. Both Jermann (1998) and Boldrin et al. (2001) lower the IES by introducing habit formation. The resulting increase in the curvature of the Bernoulli utility function, by increasing risk aversion, also takes care of increasing the volatility of the stochastic discount factor.
<unquote>

SGM; Stochastic Growth Model

EZ Two Country Recursive Formulation

https://files.nyu.edu/ht477/public/files/Tretvoll_anomaly.pdf

Asset price leads business cycles

Backus, David K.; Routledge, Bryan R.; and Zin, Stanley E., "Asset Prices in Business Cycle Analysis" (2007). Tepper School of Business. Paper 414.
http://repository.cmu.edu/cgi/viewcontent.cgi?article=1414&context=tepper

Unit Roots vs. Trend Stationary

Christiano, L. J. and M. Eichenbaum. 1990. "Unit Roots in Real Gnp: Do We Know, and Do We Care?" Carnegie-Rochester Conference Series on Public Policy

---<quote>---
Macroeconomists have traditionally viewed movements output as representing temporary fluctuations about a deterministic trend. According to this view, innovations to real gross national product (GNP) should have no impact on long-run forecasts of aggregate output. Increasingly, however, this view of aggregate fluctuations has been challenged, Following the provocative work of Nelson and Plosser (1982), numerous economists have argued that real GNP is best characterized as a stochastic process that does not revert to a deterministic trend path. Under these circumstances, innovations to real GNP should affect output forecasts into the indefinite future. In pursuing this interpretation of the data, various researchers have tried to measure the long-run response of real GNP to a shock. Estimates of this response are often referred to as the persistence of shocks to real GNP.
---<unquote>---

Trading Costs in IRBC

Backus, D. K.; P. J. Kehoe and F. E. Kydland. 1992. "International Real Business Cycles." Journal of Political Economy

---<quote>---
A surprising feature of these two experiments is that a small trading cost produces most of the properties of autarky. A possible explanation comes from Cole and Obstfeld (1991): if the gains from trade are small, then a small cost may have a large effect on the quantity of trade in goods and assets. To investigate this for our model, we measure the gains from trade by comparing equilibria in the benchmark (free-trade) economy to those in the autarky economy. We express the welfare gain as the percentage increase in the consumption path under autarky necessary to reach the same level of welfare attained with free access to international markets. Welfare in each case is estimated as the mean value of discounted utility over the 50 replications of 100 periods each. We find that consumption in autarky must be increased only 0.3 percent to make consumers as well off as they are when international markets are open. The welfare gains from trade in our theoretical economies stem solely from trade across states and dates. As in similar calculations by Cole and Obstfeld, the gains are remarkably small, which may help to account for the large effect of a small trading cost on the model's equilibria.
---<unquote>---

Complete vs. Incomplete Market (1)

Baxter, M. and M. J. Crucini. 1995. "Business Cycles and the Asset Structure of Foreign-Trade." International Economic Review

---<quote>----
The within-country correlation between saving and investment is slightly lower in the bond economy compared with the complete markets economy. This might seem surprising, since one's intuition is that closing asset markets, thus forcing individualsto bear more country-specificrisk, would act to increase within-country saving-investment correlations. However, this "basic saving measure" (defined as output minus consumption) need not be a good measure of true saving in an open economy, as discussed by Obstfeld (1986) and Stockman and Svensson (1987).
---<unquote>----

Recursive Preferences

http://www.princeton.edu/bcf/newsevents/events/lectures-in-finance/2010larshansen1.pdf

Role of EIS and Risk Aversion

http://www.lse.ac.uk/finance/prospectiveStudents/phdFinance/files09/Job_Market_Paper_Aytek_Malkhozov.pdf
Malkhozov and Samloo (2009) "Asset Prices in a News Driven Real Business Cycle Model"

<quote>
Lets first consider a Lucas-tree economy. A positive shock to expected consumption growth (or a negative shock to uncertainty) increases wealth to consumption ratio, which adjusts through movements in wealth since consumption is exogenous. This adjustment depends on the size of the elasticity of intertemporal substitution. If the substitution effect dominates the wealth effect, i.e. elasticity of intertemporal substitution is greater than one, the agent would like to hold more of the asset, thus driving prices up. Otherwise (when elasticity of intertemporal substitution is less than 1) the agent prefers bringing the increase in consumption forward, depressing prices.

How does this matter for risk premia? Shocks to expected consumption growth affect expected future returns to wealth. The agent with relative risk aversion greater than 1 wants to hedge against these changes in the investment opportunity set (and bet on them if relative risk aversion is less than one). Notice that relative risk aversion and inverse of the elasticity of intertemporal substitution are comparable measures of propensity to smooth consumption across states and time respectively. Therefore if the two are equal (CRRA case) the changes in wealth-consumption ratio exactly offset the hedging demand. With Epstein-Zin preferences there can be a wedge between relative risk aversion and the inverse of the elasticity of intertemporal substitution which will translate into premia. As an example, consider an agent with both elasticity of intertemporal substitution and relative risk aversion greater than 1, exposed to a positive shock to expected consumption growth. The intertemporal substitution effect drives up asset prices. The hedging demand effect would imply that the agent wants his portfolio to depreciate. Therefore a premium is required for the agent to hold the asset in equilibrium. If consumption and dividends are correlated the results for the pricing of aggregate risk carry forward to the risk premium for the claim on aggregate dividends. Recursive preferences are crucial for this mechanism.
<unquote>

Stochastic Volatility/ Higher-Order Approximation

http://www.aytekmalkhozov.com/papers/news_vol_asset_prices_article.pdf
see p.14

GARCH/ Stochastic Volatility Model

http://mitizane.ll.chiba-u.jp/metadb/up/AN10005358/09127216_26-3_129.pdf

http://faculty.washington.edu/ezivot/econ589/ch4.pdf

GARCH <- 単一の不確実性
SV        <- 複数の不確実性

例）リターン／ボラティティのモデル化
GARCH <- リターンへのショック（イノベーション）のみが確率過程のドライバー
SV        <- リターンとボラティリティそれぞれへのショック（イノベーション）を考慮

DGEモデルの動向

http://ir.lib.fukushima-u.ac.jp/dspace/bitstream/10270/3778/1/3-1790.pdf

外為特会の歴史

須田美矢子「外国為替資金特別会計と外国為替政策」
http://www.gakushuin.ac.jp/univ/eco/gakkai/pdf_files/keizai_ronsyuu/contents/3602/3602-33suda.pdf


渡瀬義男「外国為替資金特別会計の現状と課題」
"財務省側に次のような大前提があったことから生じたと思われる｡ すなわち､ 第一に､ 外為特会の外貨は将来の円買い介入の原資であり､ その外貨は輸出入に圧倒的比重を占めるドル建て以外にないこと､ 第二に､ ドルが暴落するような事態は当面考えられず､ 懸念される評価損が実現する事態も想定しがたいこと､ 第三に､ 貯蓄超過の日本の方が構造的に米国より金利が低いから､ 金利差逆転などは考えられないこと､ の三点である｡"

http://www.ndl.go.jp/jp/data/publication/refer/200612_671/067103.pdf

Stochastic Volatility Model

http://www2.e.u-tokyo.ac.jp/cemano/research/DP/documents/coe-j-46.pdf

GHH preferences

Greenwood-Hercowitz-Huffman(1988)AER
<quote>
Fluctuations in investment played a key role in Keynes' view of the trade cycle. There, shifts in the marginal efficiency of investment impact on investment, aggregate demand and therefore, given the disequilibrium in the labor market, employment and output. The quintessential case of this type is when there is an increase in the marginal efficiency of newly produced capital that does not affect the productivity of the capital stock already on line. When a shock of this type occurs in a standard neoclassical model, employment and output also tend to rise, but the mechanism is very different. The increase in the rate of return on investment stimulates current labor effort and output through an intertemporal substitution effect on leisure. A potential problem with this mechanism, as discussed by Robert Barro and Robert King (1984), is that intertemporal substitution which induces individuals to postpone leisure, also works to cut consumption. This effect would tend to make consumption move countercyclically, which contradicts the evidence. Labor productivity would tend to move in the " wrong" direction, too. An expansion of labor effort, given the fixed supply of capital in the short run, causes labor's productivity to decline.

In contrast to the intertemporal substitution effect mentioned above, the transmission mechanism of the investment shocks works in the present model through the optimal utilization of capital and its positive effect on the marginal productivity of labor. As will be seen, an important aspect of such a change in labor productivity is that it creates intratemporal substitution, away from leisure and toward consumption, generating procyclical effects on consumption and labor effort. Additionally, average labor productivity responds procyclically to these shocks.

That is, labor effort is determined independently of the intertemporal consumption-savings choice, which is very convenient in obtaining results from the model. As a consequence, the intertemporal substitution effect on labor effort, a central ingredient in many macroeconomic models, is eliminated. Rather than being a drawback, this implication of the utility function has the advantage of emphasizing the alternative transmission of investment shocks being studied here. When analyzing fluctuations in labor effort, this framework stresses shifts in the productivity of labor brought about by changes in the optimal rate of capacity utilization, as opposed to intertemporal substitution effects stressed by others.
<unquote>

Solow residual correlations across countries and industries

Costello(1993)JPE
<quote>
I find that aggregate output growth is correlated across countries, but aggregate productivity growth is only weakly correlated across countries. At the industry level, productivity growth is significantly correlated across industries within a country but is less correlated across countries for any individual industry. In the error-components framework, the estimated nation effects are as important as, and sometimes more important than, industry effects.

The evidence suggests that short-run productivity growth is more similar across industries in one nation than across countries for a particular industry. The results are consistent with the presence of labor hoarding if labor hoarding is truly a national occurrence. The results are also consistent with observing nation-specific technology shocks to the extent that they represent excluded factors such as human capital or the infrastructure in a country.
<unquote>

Convergence or non-convergence in output across countries

Cheung and Pascual (2004) Oxford Economic Papers
- it cannot be determined by statistical methods in common use.
- it depends on the null hypothesis of convergence or no convergence.

<quote>
The study of cross-country output dynamics from both viewpoints gives an equal opportunity for both convergence and no convergence to be validated by the data as the null hypothesis.

Our empirical results suggest that the inference about output convergence can be dictated by the choice of a null hypothesis. A conclusion of no output convergence can be reached just because no convergence is considered as the null hypothesis. Further, the no-convergence result reported in previous studies pursuing the time- series definition may be attributed to the low power of the test procedures being used. While short output data series or the use of univariate unit root procedures yields very limited support for the convergence hypothesis, the combination of long sample and efficient panel procedures delivers a more favorable result for the same hypothesis.
<unquote>

Basic Small Country IRBC Model

Mendoza(1991)AER
- They use Canadian data.
- Basic model has no adjustment costs of capital stock.
- Trade Balance is not negatively correlated with output in case relative risk aversion is high (GAMMA = 2). In case of GAMMA = 1.001, the correlation becomes negative, but the number is small. Further, if including capital adjustment costs, the correlation of them can be close to the actual data.
- Savings is not highly correlated with investments, but if including capital adjustment costs, the correlation of them can be close to the actual data.

Benchmark Model
<quote>
In general, the benchmark model is capable of mimicking the ranking of variability of the actual aggregates, and it is also consistent with some of the coefficients of autocorrelation and correlation with domestic output. Of special interest is the fact that the model mimics the absence of comovement between GDP and foreign interest payments or the trade-balance:output ratio (TB/ Y). This contrasts with the less favorable results obtained in previous empirical studies of intertemporal-equilibrium models of the current account (e.g., Ahmed, 1986; Hercowitz, 1986b).

The low correlation between S and I in the benchmark model is not related to the degree of international capital mobility. Instead, it follows from the low degree of serial autocorrelation of the shocks used to calibrate the model. With RHO = 0.36, the productivity shocks are not persistent enough to cause sufficient divergence between the expected marginal productivity of capital and the world's real interest rate to produce a stronger correlation between S and I. If, for instance, RHO is increased to 0.99, the degree of correlation between savings and investment reaches 0.8. Thus, although the benchmark model cannot mimic simultaneously the stylized facts of GDP and the correlation between savings and investment, it does support the argument presented by Obstfeld (1986) and Finn (1990), claiming that the intensity of the comovement between S and I in economies with perfect capital mobility depends on the degree of persistence of the underlying technological disturbances.
<unquote>

Adjustment Cost Model
<quote>
Perhaps the most significant result produced by these simulations is that the adjustment-cost model is capable of mimicking the two striking empirical regularities of open economies mentioned in the Introduction. Regardless of the value assigned to GAMMA, this model mimics the variability and GDP-correlation of the ratio of the trade balance to output, as well as the correlation between savings and investment. In fact, the comovement between S and I is slightly higher in both artificial economies than in the data, and this occurs without affecting the perfect international mobility of financial capital.

The introduction of moderate adjustment costs increases the persistence of the disturbances needed to calibrate the model, and with more permanent shocks investment tends to move closer together with savings, as Obstfeld (1986) suggested. Moreover, in line with the findings of Dooley et al. (1987), the perfect mobility of financial capital proves to be consistent not only with the correlation between S and I, but also with adjustment costs that prevent fast changes in physical capital.

The model mimics the variability and GDP correlation of TB/ Y because, in the presence of adjustment costs, the shocks that enable the model to mimic the stylized facts are expected to last long enough for the pro-borrowing effect, caused by an expected expansion of future output, to compensate for the pro-saving effect induced by a raise in contemporaneous output. These simulations suggest, therefore, that the intertemporal-equilibrium approach to the current account can be consistent with the empirical regularities of the business cycle.

The simulations also shed some light on a problem confronted by some empirical models of adjustment costs. As pointed out by Sargent (1978), these models generally produce reduced-form autoregressions in which highly persistent shocks cannot be distinguished from significant adjustment costs. Similarly, in the model studied here the variability of investment can be reduced by increasing the serial autocorrelation of the shocks, RHO, instead of introducing the adjustment costs. An increase in RHO reduces the probability of moving to the opposite state of productivity and lessens the chances of adjusting the capital stock, thereby reducing the variability of investment. However, the resulting persistence of the disturbances is too high and causes the model to exaggerate the actual moments. For instance, with GAMMA = 2 and PHI = 0, if RHO is set to 0.9 the variability of I falls to 5.4 percent, but the variability of GDP rises to 5 percent, and its serial autocorrelation is almost perfect. Thus, the simulations establish the relevance of adjustment costs relative to highly persistent shocks by showing that the latter are not consistent with the business cycle.
<unquote>

Appendix
<quote>
The model studied here also differs from the standard real-business-cycle prototypein its use of an endogenous rate of time preference to determine a well-defined stationary equilibrium for the holdings of foreignassets. This approach was introduced by Obstfeld (1981), following the principles formulated by Hirofumi Uzawa (1968), to analyze current-account dynamics in a deterministic model of a small open economy.
<unquote>

Habit Formation

Abel1990AER
<quote>
This paper introduces a utility function that nests three classes of utility functions: 1) time-separable utility functions; 2) "catching up with the Joneses" utility functions that depend on the consumer's level of consumption relative to the lagged cross-sectional average level of consumption; and 3) utility functions that display habit formation. Incorporating this utility function into a Lucas (1978) asset pricing model allows calculation of closed-form solutions for the prices of stocks, bills and consols under the assumption that consumption growth is i.i.d. Then equilibrium asset prices are used to examine the equity premium puzzle.

Panel C presents the unconditional expected rates of return under habit formation. The expected rates of return on both long-lived assets (stocks and consols) are extremely sensitive to the value of ALPHA (coefficient of risk averesion). Under logarithmic utility (ALPHA = 1), the expected rates of return are the same as under time-separable preferences and relative consumption. However, with ALPHA = 1.14, the expected rates of return on stocks and consols are both greater than 35 percent.
<unquote>

Catching up with the Joneses

Abel(1990)AER
<quote>
"catching up with the Joneses" utility functions that depend on the consumer's level of consumption relative to the lagged cross-sectional average level of consumption.
<unquote>

Abel(1999)JME
<quote>
The catching up with the Joneses feature of the utility function was originally
introduced to help account for the high average value of the equity premium
observed empirically. However, when this form of the utility function was
specified to imply a realistic value of the equity premium in Abel (1990), the
model produced a riskless rate of return that was far too volatile. Campbell and
Cochrane (1994) developed a form of catching up with the Joneses preferences
that yielded, as in actual data, a large equity premium and low variability of the
riskless rate. They achieved this low variability of the riskless rate by specifying
a complicated recursive function for the determination of the benchmark level of
consumption.  Here I adopt a simpler formulation of the benchmark level of
consumption that produces, with the inclusion of leverage, low variability of the
riskless rate along with a large equity premium.

The analysis of leverage arises naturally from the formulation of the canonical
asset introduced in this paper.  The payoff in period t on the canonical asset is
specified to be proportional to yt^LAMBDA, where yt is an observable random variable and LAMBDA is a constant. I use this formulation in order to include fixed-income securities and equities as special cases.

Rather than proceed with separate derivations for the prices and rates of
return for different assets such as equity, short-term bills and long-term bonds,
I will introduce a canonical asset that includes all of these assets as special cases.

The canonical asset introduced here includes equities and fixed-income securities
of all maturities.
<unquote>

Roll's critique

http://en.wikipedia.org/wiki/Roll%27s_critique

Epstein-Zin(1991)JPE
<quote>
The nominal return on the optimal portfolio is measured with the value-weighted index of shares traded on the New York Stock Exchange. A number of issues arise from the use of this measure, but the primary concern for our purposes is whether it is sufficiently broad to capture the relevant part of actual holdings of wealth; that is, Roll's (1977) critique of CAPM is relevant here. If stochastic wages are a large factor in the wealth constraint of the typical agent, then, as discussed in Section II, the return on the optimal portfolio of the agent should reflect the shadow return of the agent's human capital. Rather than attempt a lengthy analysis of this issue at this time, we shall simply assume that factors that may not be properly measured by the value- weighted index of stock returns do not affect the empirical analysis in an appreciable way. The appropriateness of this assumption, vis-a-vis the empirical results below, remains an open question.
<unquote>

Campbell(1996)JPE
<quote>
In response to the Roll (1977) critique, I extend the Campbell (1993) model to allow for human capital as a component of wealth. I impute the return on human capital from data on aggregate labor income and asset returns. Finally, I develop an econometric frame- work in which the model can be confronted with historical data.

The asset pricing model developed in Section II is empirically testable only if one can measure the return on the market portfolio. Financial economists commonly proxy the market portfolio by a value-weighted index of common stocks, but this practice is questionable. Even if the stock index return captures the return on financial wealth, as argued by Stambaugh (1982), it may not capture the return on human wealth. Approximately two-thirds of gross national product goes to labor and only one-third to capital, so human wealth is likely to be about two-thirds of total wealth and twice financial wealth. This suggests that the omission of human wealth may be a serious matter.

Increases in expected future labor income cause a positive return on human capital, but increases in expected future asset returns cause a negative return on human capital because the labor income stream is now discounted at a higher rate and is therefore worth less today.
<unquote>

Frisch Elasticity of Labor Supply

http://en.wikipedia.org/wiki/Frisch_elasticity_of_labor_supply

Elasticity of intertemporal substitution

Elasticity of intertemporal substitution (or intertemporal elasticity of substitution) is a measure of responsiveness of the growth rate of consumption to the real interest rate.

http://en.wikipedia.org/wiki/Elasticity_of_intertemporal_substitution

Principal Minors

http://home.bi.no/a0710194/Teaching/BI-Mathematics/GRA-6035/2010/lecture5-hand.pdf

Multi-Factor Model

http://web.cenet.org.cn/upfile/76975.pdf
p.10

Campbell-Shiller decomposition

http://people.bu.edu/fgourio/ec745ln_F2010.pdf

International Equity Premium Puzzle

Colacito and Croce (2010) "Risks For The Long Run And The Real Exchange Rate"

<quote>
i) risk aversion has to be large to reconcile the low volatility of consumption growth rates with highly volatile stochastic discount factors (see equity premium puzzle)

ii) consumption is poorly correlated across countries at annual or higher frequencies
<unquote>

International Equity Premium Puzzle
Since Δe = m* - m ⇨ V(Δe) = V(m*) + V(m) - 2Cov(m*,m), if Cov(m*,m) is small as is observed in the data (consumption correlation), the volatility of exchange rate in the data is too small (V(m*) and V(m) are large).

Hansen-Jagannathan Bounds

see p.12

http://people.bu.edu/fgourio/lecturenotes741Fall2006.pdf

２確率変数の分散公式

V(aX+bY) = a^2V(X)+b^2V(Y)+2abCov(X,Y)

http://mp-w3math.jwu.ac.jp/~konno/pdf/statga18a.pdf

Feldstein-Horioka Pazzle

The national saving rate is highly correlated with its domestic investment rate, which is not supported by the standard economic theory assuming open international financial economy.


Baxter and Crucini (1993) "Explaining Saving Investment Correlations"

"Saving-investment correlations are higher for larger countries."

COUNTRY SIZE AND SAVING-INVESTMENT CORRELATIONS

Country           GNP(1985 U.S. $)   S-I correlation
United States       3,994                    0.86
Japan                  1,365                     0.80
Germany                667                     0.68
France                   527                     0.31
Italy                       372                     0.39
Canada                 347                      0.61
Australia               171                      0.54
Switzerland           106                      0.65

Epstein-Zin x DSGE

https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=SED2010&paper_id=111

http://people.bu.edu/fgourio/lecturenotes741Fall2006.pdf 

http://bertha.tepper.cmu.edu/telmerc/phd_07/palgrave.pdf 

http://repository.cmu.edu/cgi/viewcontent.cgi?article=1414&context=tepper

Exchange Rates during Financial Crises

Marion Kohler (2010) "Exchange Rates during Financial Crises"
http://www.bis.org/publ/qtrpdf/r_qt1003f.pdf