Stock prices can be written as expected discounted dividends:
股票價格可以寫成預期貼現股利：

where $c$$c$cc is the dividend stream and $k$$k$kk is the discount rate. This implies that actual returns in any period are given by
其中 $c$$c$cc 是股利流， $k$$k$kk 是貼現率。這意味著任何期間的實際回報由以下公式給出

It follows (trivially) that the systematic forces that influence returns are those that change discount factors, $k$$k$kk, and expected cash flows, $E(c).{}^2$$E(c).{}^2$E(c).^(2)E(c) .{ }^{2}
由此可知（顯而易見），影響回報的系統性力量是那些改變貼現率 $k$$k$kk 和預期現金流量 $E(c).{}^2$$E(c).{}^2$E(c).^(2)E(c) .{ }^{2} 的力量。

The discount rate is an average of rates over time, and it changes with both the level of rates and the term-structure spreads across different maturities. Unanticipated changes in the riskless interest rate will therefore influence pricing, and, through their influence on the time value of future cash flows, they will influence returns. The discount rate also depends on the risk premium; hence, unanticipated changes in the premium will influence returns. On the demand side, changes in the indirect marginal utility of real wealth, perhaps as measured by real consumption changes, will influence pricing, and such effects should also show up as unanticipated changes in risk premia.
貼現率是一段時間內利率的平均值，它會隨著不同期限的利率水平和期限結構差異而改變。因此，無風險利率的意外變化會影響定價，並透過對未來現金流量時間值的影響，影響回報。貼現率也取決於風險溢價；因此，溢價的非預期變化會影響回報。在需求方面，實際財富的間接邊際效用的變化，或許可以用實際消費變化來衡量，將影響定價，而這種影響也應該顯示為風險溢價的非預期變化。

Expected cash flows change because of both real and nominal forces. Changes in the expected rate of inflation would influence nominal expected cash flows as well as the nominal rate of interest. To the extent that pricing is done in real terms, unanticipated price-level changes will have a systematic effect, and to the extent that relative prices change along with general inflation, there can also be a change in asset valuation associated with changes in the average inflation rate. Finally, changes in the expected level of real production would affect the current real value of cash flows. Insofar as the risk-premium measure does not capture industrial production uncertainty, innovations in the rate of productive activity should have an influence on stock returns through their impact on cash flows.
預期現金流量會因為實際和名目兩種力量而改變。預期通貨膨脹率的變化會影響名義預期現金流量以及名義利率。在以實質價格定價時，未預期的價格水平變化會有系統性的影響，而在相對價格隨著一般通貨膨脹而變化時，也會出現與平均通貨膨脹率變化相關的資產估值變化。最後，預期實際生產水平的變化會影響現金流量的當前實際價值。由於風險溢價計量並未反映工業生產的不確定性，生產活動率的創新應該會透過其對現金流量的影響而對股票回報產生影響。

Having proposed a set of relevant variables, we must now specify their measurement and obtain time series of unanticipated movements. We could proceed by identifying and estimating a vector autoregressive model in an attempt to use its residuals as the unanticipated innova-
在提出一組相關變數之後，我們現在必須指定這些變數的量測方式，並取得非預期變動的時間序列。我們可以識別並估計一個向量自回歸模型，嘗試使用其殘值作為非預期的創新。

tions in the economic factors. It is, however, more interesting and (perhaps) robust out of sample to employ theory to find single equations that can be estimated directly. In particular, since monthly rates of return are nearly serially uncorrelated, they can be employed as innovations without alteration. The general impact of a failure adequately to filter out the expected movement in an independent variable is to introduce an errors-in-variables problem. This has to be traded off against the error introduced by misspecification of the estimated equation for determining the expected movement.
經濟因素的影響。然而，在樣本之外，運用理論來尋找可直接估算的單一方程式會更有趣且（或許）更穩 健。特別是，由於每月的回報率幾乎是無序列關聯的，因此可以不加改動地將其作為創新。如果無法充分濾除自變數的預期變動，一般的影響就是會產生變數誤差的問題。這個問題必須與決定預期變動的估算等式的錯誤定義所引入的誤差相抵銷。

A somewhat subtler version of the same problem arises with procedures such as vector autoregression. Any such statistically based timeseries approach will find lagged stock market returns having a significant predictive content for macroeconomic variables. In the analysis of pricing, then, we will indirectly be using lagged stock market variables to explain the expected returns on portfolios of stocks. Whatever econometric advantages such an approach might offer, it is antithetical to the spirit of this investigation, which is to explore the pricing influence of exogenous macroeconomic variables. For this reason, as much as for any other, we have chosen to follow the simpler route in constructing the time series we use. ${}^3$${}^3$^(3){ }^{3}
向量自回歸(vector autoregression)等程序會產生相同問題的較微妙版本。任何此類以統計學為基礎的時序方法，都會發現滯後的股市回報對宏觀經濟變數有顯著的預測作用。因此，在定價分析中，我們會間接使用滯後的股市變數來解釋股票組合的預期回報。無論這種方法在計量經濟學上有什麼優點，它都與本研究的精神背道而馳，因為本研究的精神是探討外生的宏觀經濟變數對於定價的影響。基於這個原因，我們選擇了更簡單的方法來建立我們使用的時間序列。 ${}^3$${}^3$^(3){ }^{3}

Throughout this paper we adopt the convention that time subscripts apply to the end of the time period. The standard period is 1 month. Thus, $E(\mid t-1)$$E(\mid t-1)$E(∣t-1)E(\mid t-1) denotes the expectation operator at the end of month $t-1$$t-1$t-1t-1 conditional on the information set available at the end of month $t$$t$tt -1 , and $X(t)$$X(t)$X(t)X(t) denotes the value of variable $X$$X$XX in month $t$$t$tt, or the growth that prevailed from the end of $t-1$$t-1$t-1t-1 to the end of $t$$t$tt.
在本文中，我們採用時間下標適用於時期結束的慣例。標準期間為 1 個月。因此， $E(\mid t-1)$$E(\mid t-1)$E(∣t-1)E(\mid t-1) 表示 $t-1$$t-1$t-1t-1 月底的期望算子，條件是 $t$$t$tt -1 月底的可用資訊集，而 $X(t)$$X(t)$X(t)X(t) 表示變數 $X$$X$XX 在 $t$$t$tt 月的值，或從 $t-1$$t-1$t-1t-1 月底到 $t$$t$tt 月底的普遍成長。

The basic series is the growth rate in U.S. industrial production. It was obtained from the Survey of Current Business. If IP $(t)$$(t)$(t)(t) denotes the rate of industrial production in month $t$$t$tt, then the monthly growth rate is
基本數列是美國工業生產的成長率。它取自於「當前業務調查」（Survey of Current Business）。如果 IP $(t)$$(t)$(t)(t) 表示 $t$$t$tt 月份的工業生產率，則每月的增長率為

and the yearly growth rate is
而每年的成長率為

(see table 1 for a summary of variables).
(變數摘要請參閱表 1）。
Because $\mathrm{IP}(t)$$\mathrm{IP}(t)$IP(t)\operatorname{IP}(t) actually is the flow of industrial production during month $t,\mathrm{MP}(t)$$t,\mathrm{MP}(t)$t,MP(t)t, \operatorname{MP}(t) measures the change in industrial production lagged by at least a partial month. To make this variable contemporaneous with other series, subsequent statistical work will lead it by 1 month. Except for an annual seasonal, it is noisy enough to be treated as an innovation.
由於 $\mathrm{IP}(t)$$\mathrm{IP}(t)$IP(t)\operatorname{IP}(t) 實際上是當月的工業生產流量，因此 $t,\mathrm{MP}(t)$$t,\mathrm{MP}(t)$t,MP(t)t, \operatorname{MP}(t) 量度的是至少滯後一個月的工業生產變化。為了讓這個變數與其他序列同步，後續的統計工作會將它滯後 1 個月。除了年度季節性之外，它的雜訊足以被視為創新。
3. In addition, the pricing tests reported below used portfolios that have induced autocorrelations in their returns arising from the nontrading effect.
3.此外，下文報告的定價測試所使用的投資組合，其回報會因非交易效應而產生誘發自相關性。

TABLE $1$$1$1quad1 \quad Glossary and Definitions of Variables
表 $1$$1$1quad1 \quad 變數的詞彙與定義

The monthly series of yearly growth rates, $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t), was examined because the equity market is related to changes in industrial activity in the long run. Since stock market prices involve the valuation of cash flows over long periods in the future, monthly stock returns may not be highly related to contemporaneous monthly changes in rates of industrial production, although such changes might capture the information pertinent for pricing. This month’s change in stock prices probably reflects changes in industrial production anticipated many months into
研究年增長率的每月序列 $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t) ，是因為股票市場與長期的工業活動變化有關。由於股票市場價格涉及未來長期現金流量的估值，因此每月股票回報可能與同期工業生產率的每月變化關係不大，儘管這些變化可能捕捉到與定價相關的資訊。本月的股票價格變化可能反映了未來多個月的工業生產預測變化。
the future. Therefore, subsequent statistical work will lead this variable by 1 year, similar to the variable used in Fama (1981).
未來。因此，後續的統計工作會將此變數導向 1 年，類似於 Fama (1981) 所使用的變數。

Because of the overlap in the series, $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t) is highly autocorrelated. A procedure was developed for forecasting expected $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t) and a series of unanticipated changes in $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t), and changes in the expectation itself were examined for their influence on pricing. The resulting series offered no discernible advantage over the raw production series, and, as a consequence, they have been dropped from the analysis. ${}^4$${}^4$^(4){ }^{4}
由於序列的重疊， $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t) 具有高度的自關性。我們開發了一套程序來預測預期 $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t) 和 $\mathrm{YP}(t)$$\mathrm{YP}(t)$YP(t)\mathrm{YP}(t) 的一系列非預期變化，並檢查預期本身的變化對定價的影響。由此產生的系列與原始生產系列相比沒有明顯的優勢，因此，這些系列已從分析中剔除。 ${}^4$${}^4$^(4){ }^{4}

Unanticipated inflation is defined as
非預期通貨膨脹的定義為

where $\mathrm{I}(t)$$\mathrm{I}(t)$I(t)\mathrm{I}(t) is the realized monthly first difference in the logarithm of the Consumer Price Index for period $t$$t$tt. The series of expected inflation, $\mathrm{E}[\mathrm{I}(t)\mid t-1]$$\mathrm{E}[\mathrm{I}(t)\mid t-1]$E[I(t)∣t-1]\mathrm{E}[\mathrm{I}(t) \mid t-1] for the period 1953-78, is obtained from Fama and Gibbons (1984). If RHO $(t)$$(t)$(t)(t) denotes the ex post real rate of interest applicable in period $t$$t$tt and $\mathrm{TB}(t-1)$$\mathrm{TB}(t-1)$TB(t-1)\mathrm{TB}(t-1) denotes the Treasury-bill rate known at the end of period $t-1$$t-1$t-1t-1 and applying to period $t$$t$tt, then Fisher’s equation asserts that
其中 $\mathrm{I}(t)$$\mathrm{I}(t)$I(t)\mathrm{I}(t) 是 $t$$t$tt 期間消費者物價指數對數的已實現每月第一差值。1953-78 年間的預期通貨膨脹系列 $\mathrm{E}[\mathrm{I}(t)\mid t-1]$$\mathrm{E}[\mathrm{I}(t)\mid t-1]$E[I(t)∣t-1]\mathrm{E}[\mathrm{I}(t) \mid t-1] 取自 Fama and Gibbons (1984)。如果 RHO $(t)$$(t)$(t)(t) 表示在 $t$$t$tt 期間適用的事後實際利率，而 $\mathrm{TB}(t-1)$$\mathrm{TB}(t-1)$TB(t-1)\mathrm{TB}(t-1) 表示在 $t-1$$t-1$t-1t-1 期間結束時已知並適用於 $t$$t$tt 期間的國庫券利率，那麼費雪公式會斷言

Hence, $\mathrm{TB}(t-1)-\mathrm{I}(t)$$\mathrm{TB}(t-1)-\mathrm{I}(t)$TB(t-1)-I(t)\mathrm{TB}(t-1)-\mathrm{I}(t) measures the ex post real return on Treasury bills in the period. From a time-series analysis of this variable, Fama and Gibbons (1984) constructed a time series for $\mathrm{E}[\mathrm{RHO}(t)\mid t-1]$$\mathrm{E}[\mathrm{RHO}(t)\mid t-1]$E[RHO(t)∣t-1]\mathrm{E}[\mathrm{RHO}(t) \mid t-1]. Our expected inflation variable is defined by subtracting their time series for the expected real rate from the $\mathrm{TB}(t-1)$$\mathrm{TB}(t-1)$TB(t-1)\mathrm{TB}(t-1) series.
因此， $\mathrm{TB}(t-1)-\mathrm{I}(t)$$\mathrm{TB}(t-1)-\mathrm{I}(t)$TB(t-1)-I(t)\mathrm{TB}(t-1)-\mathrm{I}(t) 衡量的是該期間國庫券的事後實際回報。Fama 和 Gibbons (1984) 從這個變數的時間序列分析中，建構了 $\mathrm{E}[\mathrm{RHO}(t)\mid t-1]$$\mathrm{E}[\mathrm{RHO}(t)\mid t-1]$E[RHO(t)∣t-1]\mathrm{E}[\mathrm{RHO}(t) \mid t-1] 的時間序列。我們的預期通貨膨脹變數的定義是從 $\mathrm{TB}(t-1)$$\mathrm{TB}(t-1)$TB(t-1)\mathrm{TB}(t-1) 系列中減去他們的預期實際利率時間序列。

Another inflation variable that is unanticipated and that might have an influence separable from UI is
另一個未預期的通貨膨脹變數，其影響可能與 UI 分離，那就是

the change in expected inflation. We subscript this variable with $t$$t$tt since it is (in principle) unknown at the end of month $t-1$$t-1$t-1t-1. While, strictly speaking, DEI $(t)$$(t)$(t)(t) need not have mean zero, under the additional assumption that expected inflation follows a martingale this variable may be treated as an innovation, and it may contain information not present in the UI variable. This would occur whenever inflation forecasts are influenced by economic factors other than past forecasting errors. (Notice that the UI series and the DEI series will contain the information in a series of innovations in the nominal interest rate, TB. ${)}^5$${)}^5$)^(5))^{5}
預期通貨膨脹的變化。我們用 $t$$t$tt 來標記這個變數，因為它在 $t-1$$t-1$t-1t-1 月底（原則上）是未知的。雖然嚴格來說，DEI $(t)$$(t)$(t)(t) 的均值不一定為零，但在預期通貨膨脹遵循馬丁式（martingale）的額外假設下，這個變數可被視為創新，它可能包含 UI 變數中沒有的資訊。當通貨膨脹預測受到過去預測錯誤以外的經濟因素影響時，就會出現這種情況。(請注意，UI 數列和 DEI 數列將包含名目利率 TB 的一系列創新中的資訊。 ${)}^5$${)}^5$)^(5))^{5}

To capture the effect on returns of unanticipated changes in risk premia, we will employ another variable drawn from the money markets. The variable, UPR, is defined as
為了捕捉風險溢價的非預期變化對回報的影響，我們將採用另一個來自貨幣市場的變數。變數 UPR 定義為
$\mathrm{UPR}(t)=$$\mathrm{UPR}(t)=$UPR(t)=\operatorname{UPR}(t)= ‘Baa and under’’ bond portfolio return $(t)-\mathrm{LGB}(t)$$(t)-\mathrm{LGB}(t)$(t)-LGB(t)(t)-\operatorname{LGB}(t),
$\mathrm{UPR}(t)=$$\mathrm{UPR}(t)=$UPR(t)=\operatorname{UPR}(t)= 「Baa 及以下」' 債券組合回報 $(t)-\mathrm{LGB}(t)$$(t)-\mathrm{LGB}(t)$(t)-LGB(t)(t)-\operatorname{LGB}(t) 、
where $\mathrm{LGB}(t)$$\mathrm{LGB}(t)$LGB(t)\operatorname{LGB}(t) is the return on a portfolio of long-term government bonds obtained from Ibbotson and Sinquefield (1982) for the period 1953-78. From 1979 through 1983, LGB $(t)$$(t)$(t)(t) was obtained from the Center for Research in Securities Prices (CRSP) data file. Again, UPR is not formally an innovation, but, as the differences in two return series, it is sufficiently uncorrelated that we can treat it as unanticipated, and we will use it as a member of the set of economic factors.
其中， $\mathrm{LGB}(t)$$\mathrm{LGB}(t)$LGB(t)\operatorname{LGB}(t) 是長期政府債券組合的報酬率，取自 Ibbotson 和 Sinquefield (1982)，期間為 1953-78 年。從 1979 年到 1983 年，LGB $(t)$$(t)$(t)(t) 取自證券價格研究中心 (CRSP) 的資料檔案。同樣地，UPR 並非正式的創新，但由於它是兩個回報序列的差異，具有足夠的非相關性，我們 可以將它視為非預期的，因此我們將它作為經濟因素集的一員。

The low-grade bond return series is for nonconvertible corporate bonds, and it was obtained from R. G. Ibbotson and Company for the period prior to 1977. A detailed description of the sample is contained in Ibbotson (1979). The low-grade series was extended through 1983 by choosing 10 bonds whose ratings on January 1966 were below Baa. By 1978 these bonds still were rated below Baa, but their maturity was shorter than that of the long-term government bond series. These 10 bonds were then combined with three that were left over from the Ibbotson series at the end of 1978 to create a low-grade bond portfolio of 13 bonds in all. The returns on this portfolio were then used to extend the UPR series beyond 1977 and through 1983. Two further difficulties with the series are that the ratings have experienced considerable inflation since the mid-1950s and that the low-grade series contains bonds that are unrated.
低等級債券收益率系列為不可轉換公司債券，1977 年以前的數據來自 R. G. Ibbotson and Company。Ibbotson (1979)對於樣本的詳細說明。低評級系列透過選擇 1966 年 1 月評等低於 Baa 的 10 筆債券，延伸至 1983 年。到 1978 年，這些債券的評等仍低於 Baa，但其到期日短於長期政府債券系列。這 10 種債券再加上 1978 年底 Ibbotson 系列剩下的 3 種債券，就形成了總共 13 種債券的低等級債券組合。這個組合的回報被用來將 UPR 系列延伸至 1977 年以後，直到 1983 年。該系列還有兩個難題，一是自 1950 年代中期以來，評等經歷了相當大的通貨膨脹，二是低等級系列包含了未評等級的債券。

The UPR variable would have mean zero in a risk-neutral world, and it is natural to think of it as a direct measure of the degree of risk aversion implicit in pricing (at least insofar as the rating agencies maintain constant standards for their classifications). We hoped that UPR would reflect much of the unanticipated movement in the degree of risk aversion and in the level of risk implicit in the market’s pricing of stocks. ${}^6$${}^6$^(6){ }^{6}
在風險中性的世界中，UPR 變數的平均值為零，因此很自然地將其視為直接量度定價中隱含的風險迴避程度 (至少在評等機構維持其分類標準不變的情況下是如此)。我們希望 UPR 能夠反映出風險厭惡程度和市場股票定價中隱含的風險水平的大部分意外變動。 ${}^6$${}^6$^(6){ }^{6}

To capture the influence of the shape of the term structure, we employ another interest rate variable,
為了捕捉期限結構形狀的影響，我們採用了另一個利率變數、

Again, under the appropriate form of risk neutrality,
同樣，在風險中性的適當形式下、

and this variable can be thought of as measuring the unanticipated return on long bonds. The assumption of risk neutrality is used only to isolate the pure term-structure effects; the variable UPR is used to capture the effect of changes in risk aversion.
而此變數可被視為量度長期債券的非預期回報。風險中性的假設只用來隔離純粹的期限結構效應；變數 UPR 用來捕捉風險厭惡的變化效應。

The major thrust of our effort is to examine the relation between nonequity economic variables and stock returns. However, because of the smoothing and averaging characteristics of most macroeconomic time series, in short holding periods, such as a single month, these series cannot be expected to capture all the information available to the market in the same period. Stock prices, on the other hand, respond very quickly to public information. The effect of this is to guarantee that market returns will be, at best, weakly related and very noisy relative to innovations in macroeconomic factors.
我們努力的重點在於檢視非股票經濟變數與股票回報之間的關係。然而，由於大多數宏觀經濟時間序列都具有平滑化和平均化的特性，因此在短持有期（例如單月）內，我們無法期望這些序列能夠捕捉到市場在同一時期內可獲得的所有資訊。另一方面，股票價格對公開資訊的反應非常迅速。這就保證了市場回報充其量只是與宏觀經濟因素的創新呈現微弱的關係，而且非常嘈雜。

This should bias our results in favor of finding a stronger linkage between the time-series returns on market indices and other portfolios of stock returns than between these portfolio returns and innovations in the macro variables. To examine the relative pricing influence of the traditional market indices we used the following variables:
這應會使我們的結果偏向於發現市場指數的時間序列回報與其他股票回報組合之間的聯繫，比這些組合回報與宏觀變數的創新之間的聯繫更強。為了檢視傳統市場指數的相對定價影響，我們使用了下列變數：
$\mathrm{EWNY}(t)=$$\mathrm{EWNY}(t)=$EWNY(t)=\operatorname{EWNY}(t)= return on the equally weighted NYSE index;
$\mathrm{EWNY}(t)=$$\mathrm{EWNY}(t)=$EWNY(t)=\operatorname{EWNY}(t)= 等權紐約證券交易所指數的報酬率；
$\mathrm{VWNY}(t)=$$\mathrm{VWNY}(t)=$VWNY(t)=\operatorname{VWNY}(t)= return on the value-weighted NYSE index.
$\mathrm{VWNY}(t)=$$\mathrm{VWNY}(t)=$VWNY(t)=\operatorname{VWNY}(t)= 價值加權 NYSE 指數的報酬率。
These variables should reflect both the real information in the industrial production series and the nominal influence of the inflation variables.
這些變數應同時反映出工業生產系列的實際資訊，以及通貨膨脹變數的名義影響。

In addition to the macro variables discussed above, we also examined a time series of percentage changes in real consumption, CG. The series is in real per capita terms and includes service flows. It was constructed by dividing the CITIBASE series of seasonally adjusted real consumption (excluding durables) by the Bureau of Census’s monthly population estimates. The CG series extends from January 1959 to December 1983, and it is an extension of a series obtained from Lars Hansen for the period through 1979. A detailed description of its construction can be found in Hansen and Singleton (1983).
除了以上討論的宏觀變數外，我們也檢視了實際消費百分比變化的時間序列 CG。該數列以實際人均計算，並包括服務流量。它是由 CITIBASE 經季節性調整的實際消費（不含耐用品）數列除以人口普查局的每月人口估計數 字所組成。CG 系列從 1959 年 1 月延伸至 1983 年 12 月，是 Lars Hansen 所提供的 1979 年以前的系列的延伸。有關其結構的詳細說明，請參閱 Hansen and Singleton (1983)。

It is often argued that oil prices must be included in any list of the systematic factors that influence stock market returns and pricing. To test this proposition and to examine another alternative to the macro variables discussed above, we formed the OG series of realized
常有人認為，在任何影響股市回報和定價的系統因素清單中，都必須包括油價。為了驗證這一主張，並研究上述宏觀變數的另一種替代方案，我們將已實現的石油價格形成 OG 系 列。
monthly first differences in the logarithm of the Producer Price Index/ Crude Petroleum series (obtained from the Bureau of Labor Statistics, U.S. Department of Labor, DRI series no. 3884). The glossary in table 1 summarizes the variables.
生產者價格指數/原油系列對數的每月第一次差異（來自美國勞工部勞工統計局，DRI 系列 No.3884).表 1 的詞彙總結了變數。

Table 2 displays the correlation matrix for the state variables. The correlation matrices of table 2 are computed for several different pe-
表 2 顯示狀態變數的相關矩陣。表 2 中的相關矩陣是針對幾個不同的狀態變數計算出來的。

TABLE 2 Correlation Matrices for Economic Variables
表 2 經濟變數的相關矩陣

Note.-VWNY = return on the value-weighted NYSE index; EWNY = return on the equally weighted NYSE index; MP = monthly growth rate in industrial production; DEI = change in expected inflation; UI = unanticipated inflation; UPR = unanticipated change in the risk premium (Baa and under return - long-term government bond return); UTS = unanticipated change in the term structure (long-term government bond return - Treasury-bill rate); and YP = yearly growth rate in industrial production.
註：-VWNY = 紐約證交所價值加權指數回報；EWNY = 紐約證交所等權加權指數回報；MP = 工業生產月度成長率；DEI = 預期通貨膨脹變化；UI = 未預期通貨膨脹；UPR = 未預期風險溢價變化 (Baa 及以下回報 - 長期政府債券回報)；UTS = 未預期期限結構變化 (長期政府債券回報 - 國庫債券利率)；YP = 工業生產年度成長率。
TABLE 3 Autocorrelations of the Economic Variables, January 1953-November 1983
表 3 經濟變數的自相關性，1953 年 1 月至 1983 年 11 月

NoTE.-YP = yearly growth rate in industrial production; $\mathrm{MP}=$$\mathrm{MP}=$MP=\mathrm{MP}= monthly growth rate in industrial production; DEI $=$$=$== change in expected inflation; UI $=$$=$== unanticipated inflation; UPR = unanticipated change in the risk premium (Baa and under return - long-term government bond return); UTS = unanticipated change in the term structure (long-term government bond return - Treasury-bill rate); ${\mathrm{CG}}_{\text{F}}=$${\mathrm{CG}}_{\text{F }}=$CG_("F ")=\mathrm{CG}_{\text {F }}= growth rate in real per capita consumption; $\mathrm{OG}=$$\mathrm{OG}=$OG=\mathrm{OG}= growth rate in oil prices; EWNY $=$$=$== return on the equally weighted NYSE index; and VWNY = return on the value-weighted NYSE index.
NoTE.-YP = 工業生產年增長率； $\mathrm{MP}=$$\mathrm{MP}=$MP=\mathrm{MP}= 工業生產月增長率；DEI $=$$=$== 預期通貨膨脹變化；UI $=$$=$== 未預期通貨膨脹；UPR = 未預期風險溢價變化（Baa 及以下回報率 - 長期政府債券回報率）；UTS = 期限結構的非預期變化（長期政府債券回報率 - 國庫券利率）； ${\mathrm{CG}}_{\text{F}}=$${\mathrm{CG}}_{\text{F }}=$CG_("F ")=\mathrm{CG}_{\text {F }}= 實際人均消費成長率； $\mathrm{OG}=$$\mathrm{OG}=$OG=\mathrm{OG}= 油價成長率；EWNY $=$$=$== 等權 NYSE 指數的回報率；以及 VWNY = 價值加權 NYSE 指數的回報率。
riods; part A covers the entire 371-month sample period from January 1953 through November 1983, and the remaining parts cover three subperiods, with breaks at December 1977 and January 1973. We have broken the sample at this time because it is often argued that the oil price jump in 1973 presaged a structural shift in the macro variables. (The work of Litterman and Weiss [1983] supports this view, but, although we have performed no formal tests, the correlation matrix does not appear to differ markedly across the subperiods.)
A 部分涵蓋從 1953 年 1 月到 1983 年 11 月的 371 個月樣本期間，其餘部分涵蓋三個子期間，並在 1977 年 12 月和 1973 年 1 月中斷。我們在這個時期中斷樣本，是因為有人認為 1973 年的油價跳升預示了宏觀變數的結構轉變。(Litterman 和 Weiss [1983] 的研究支持這個觀點，不過，雖然我們沒有進行正式的測試，但相關矩陣在子期間似乎沒有明顯的差異）。

With the exception of the market indices, the strongest correlation is between UPR and UTS. This is to be expected since they both use the long-term bond series, $\mathrm{LGB}(t)$$\mathrm{LGB}(t)$LGB(t)\operatorname{LGB}(t). The resulting collinearity tends to weaken the individual impact of these variables. Substituting an Aaa corporate bond series for treasuries in the definition of UPR did, in fact, improve the significance of both UPR and UTS, but the improvement was not sufficiently important to make a qualitative difference in our findings.
除市場指數外，UPR 與 UTS 的相關性最高。這是可以預期的，因為它們都使用長期債券系列 $\mathrm{LGB}(t)$$\mathrm{LGB}(t)$LGB(t)\operatorname{LGB}(t) 。由此產生的共線性傾向於削弱這些變量的個別影響。在 UPR 的定義中，以 Aaa 公司債券系列取代國庫債券，事實上確實改善了 UPR 和 UTS 的顯著性，但改善的程度不足以使我們的研究結果產生質性的差異。
The production series, YP and MP, are correlated with each other and with each of the other variables except DEI and UI, which are also strongly correlated. These latter two series are correlated because they both contain the $\mathrm{EI}(t)$$\mathrm{EI}(t)$EI(t)\mathrm{EI}(t) series, and the negative correlation between DEI and UTS occurs for a similar reason. A number of other correlations are not negligible, but the variables are far from perfectly correlated, and no one variable can be substituted for any other.
除了 DEI 和 UI 也是強烈相關之外，YP 和 MP 這兩個生產序列彼此相關，也與其他每個變數相關。這後兩個數列之所以相關，是因為它們都包含 $\mathrm{EI}(t)$$\mathrm{EI}(t)$EI(t)\mathrm{EI}(t) 數列，而 DEI 與 UTS 之間出現負相關也是基於類似原因。其他一些相關性也不容忽視，但這些變數遠非完全相關，沒有任何一個變數可以取代其他變數。

Table 3 displays the autocorrelations for the state variables computed over the entire sample period, January 1953-November 1983. There are no surprises here; as expected, YP is highly autocorrelated. The variables generally display mild autocorrelations, and many of them have seasonals at the 12 -month lag. The MP series, in particular, has a peak in its lag at 12 months (repeated at 24 months), warning that this variable is highly seasonal. As noted above, the autocorrelation in the state variables implies the existence of an errors-in-variables problem that will bias estimates of the loadings of the stock returns on these variables and will bias downward estimates of statistical significance.
表 3 顯示 1953 年 1 月至 1983 年 11 月整個樣本期間所計算的狀態變數自相關性。表 3 顯示 1953 年 1 月至 1983 年 11 月整個樣本期間國家變數的自關係。這些變數一般都顯示出輕微的自相關，而且許多變數在 12 個月的滯後期都有季節性。特別是 MP 數列，在 12 個月的滯後期有一個峰值（在 24 個月的滯後期重複出現），警示這個變數具有高度的季節性。如上所述，狀態變數的自相關性意味著存在變數誤差的問題，這將使股票回報在這些變數 上的負載估計出現偏差，並使統計顯著性的向下估計出現偏差。

Using the state variables ${}^7$${}^7$^(7){ }^{7} defined above implies that individual stock returns follow a factor model of the form
使用上述定義的狀態變數 ${}^7$${}^7$^(7){ }^{7} ，意味著個股回報會遵循以下形式的因子模型

[

$$\begin{array}{rl}R=& a+b_{\mathrm{MP}}\mathrm{MP}+b_{\mathrm{DEI}}\mathrm{DEI}+b_{\mathrm{UI}}\mathrm{UI}\\ & +b_{\mathrm{UPR}}\mathrm{UPR}+b_{\mathrm{UTS}}\mathrm{UTS}+e\end{array}$$$$\begin{array}{rl}R=& a+b_{\mathrm{MP}}\mathrm{MP}+b_{\mathrm{DEI}}\mathrm{DEI}+b_{\mathrm{UI}}\mathrm{UI}\\ & +b_{\mathrm{UPR}}\mathrm{UPR}+b_{\mathrm{UTS}}\mathrm{UTS}+e\end{array}$${:[R=a+b_(MP)MP+b_(DEI)DEI+b_(UI)UI],[+b_(UPR)UPR+b_(UTS)UTS+e]:}\begin{aligned} R= & a+b_{\mathrm{MP}} \mathrm{MP}+b_{\mathrm{DEI}} \mathrm{DEI}+b_{\mathrm{UI}} \mathrm{UI} \\ & +b_{\mathrm{UPR}} \mathrm{UPR}+b_{\mathrm{UTS}} \mathrm{UTS}+e \end{aligned}
]

where the betas are the loadings on the state variables, $a$$a$aa is the constant term, and $e$$e$ee is an idiosyncratic error term. To ascertain whether the identified economic state variables are related to the underlying factors that explain pricing in the stock market, a version of the FamaMacBeth (1973) technique was employed. The procedure was as follows. (a) A sample of assets was chosen. (b) The assets’ exposure to the economic state variables was estimated by regressing their returns on the unanticipated changes in the economic variables over some estimation period (we used the previous 5 years). © The resulting estimates of exposure (betas) were used as the independent variables in 12 cross-sectional regressions, one regression for each of the next 12 months, with asset returns for the month being the dependent variable. Each coefficient from a cross-sectional regression provides an estimate of the sum of the risk premium, if any, associated with the state variable and the unanticipated movement in the state variable for that month. (d) Steps $b$$b$bb and $c$$c$cc were then repeated for each year in the sample, yielding for each macro variable a time series of estimates of its associated risk premium. The time-series means of these estimates were then tested by a $t$$t$tt-test for significant difference from zero.
其中，betas 是狀態變數的負載， $a$$a$aa 是常數項， $e$$e$ee 是特異誤差項。為了確定已識別的經濟狀態變數是否與解釋股票市場定價的基本因素有關，我們採用了 FamaMacBeth (1973) 技術的版本。程序如下(a) 選擇資產樣本。(b) 將資產的報酬率與某個估計期間（我們採用前 5 年）經濟變數的非預期變化作回歸，以估 計資產對經濟狀態變數的風險。由此估算出的風險值（betas）在 12 次橫截面迴歸中作為自變數使用，未來 12 個月各進行一次迴歸，以該月的資產報酬率作為因變數。截面迴歸的每個係數都提供與狀態變數相關的風險溢價（如有）與該月狀態變數的非預期變動之和的估計值。(d) 然後，對於樣本中的每一年，重複步驟 $b$$b$bb 和 $c$$c$cc ，得出每個宏觀變數的相關風險溢價估計的時間序列。這些估算值的時間序列均值隨後進行 $t$$t$tt 檢驗，以確定是否與零有顯著差異。

To control the errors-in-variables problem that arises from the use at step $c$$c$cc of the beta estimates obtained in step $b$$b$bb and to reduce the noise in individual asset returns, the securities were grouped into portfolios. An effort was made to construct the portfolios so as to spread their expected returns over a wide range in an effort to improve the discriminatory power of the cross-sectional regression tests. To accomplish this spreading we formed portfolios on the basis of firm.size. Firm size is known to be strongly related to average return (see Banz 1981), and we hoped that it would provide the desired dispersion without biasing the tests of the economic variables. (It has been facetiously noted that size may be the best theory we now have of expected returns. Unfortunately, this is less of a theory than an empirical observation. ${)}^8$${)}^8$)^(8))^{8}
為了控制由於在步驟 $c$$c$cc 中使用在步驟 $b$$b$bb 中取得的貝他係數估計所產生的變數誤差問題，並減少個別資產回報的雜訊，我們將證券組合為投資組合。為了提高截面回歸測試的判別力，我們努力建構投資組合，使其預期回報在廣泛的範圍內分散。為了達到此目的，我們以公司規模為基礎來組成投資組合。眾所周知，公司規模與平均報酬率關係密切（見 Banz 1981），我們希望它能提供所需的分散性，而不會對經濟變數的測試造成偏差。(有人揶揄地指出，規模可能是我們目前對預期回報率的最佳理論。不幸的是，與其說這是一種理論，不如說這是一種經驗觀察。 ${)}^8$${)}^8$)^(8))^{8}

Table 4 reports the results of these tests on 20 equally weighted portfolios, grouped according to the total market values of their constituent securities at the beginning of each test period. Each part of table 4 is broken into four subperiods beginning with January 1958, the first month preceded by the requisite 60 months of data used to estimate exposures. Part A of table 4 examines the state variables, YP, MP, DEI, UI, UPR, and UTS. Over the entire sample period MP, UI, and UPR are significant, while UTS is marginally so. The inflationrelated variables, DEI and UI, were highly significant in the 1968-77 period and insignificant both earlier and later. The yearly production series, YP, was not significant in any subperiod, and, as can be seen from part B, deleting it had no substantive effect on the remaining state variables. Although the coefficients have the same signs as in the overall period, they are generally smaller in absolute magnitude and less significant in the last subperiod, 1978-84. ${}^9$${}^9$^(9){ }^{9}
表 4 匯報了 20 個等權重投資組合的測試結果，這些組合是根據每個測試期間開始時的成 分證券總市值來分類的。表 4 的每個部分從 1958 年 1 月開始分成四個子期間，1958 年 1 月是估計風險所需的 60 個月資料之前的第一個月。表 4 的 A 部分檢視了狀態變數 YP、MP、DEI、UI、UPR 和 UTS。在整個樣本期間，MP、UI 和 UPR 都是顯著的，而 UTS 則略微顯著。與通貨膨脹相關的變數 DEI 和 UI 在 1968-77 年間非常顯著，但在之前和之後都不顯著。年產量序列 YP 在任何子期間都不顯著，而且從 B 部分可以看出，刪除 YP 對其餘的國家變數沒有實質影響。雖然這些係數的符號與整體期間的相同，但在 1978-84 年這個最後的次期間，這些係數的絕對值通常較小，也較不顯著。 ${}^9$${}^9$^(9){ }^{9}
While we have not developed a theoretical foundation for the signs of the state variables, it is worth noting that their signs are, at least, plausible. The positive sign on MP reflects the value of insuring against real systematic production risks. Similarly, UPR has a positive risk premium since individuals would want to hedge against unanticipated increases in the aggregate risk premium occasioned by an increase in uncertainty. Since changes in inflation have the general effect of shifting wealth among investors, there is no strong a priori presumption that would sign the risk premia for UI or DEI, but the negative signs on the premia for these variables probably mean that stock market assets are generally perceived to be hedges against the adverse influence on other assets that are, presumably, relatively more fixed in nominal terms.
儘管我們沒有為狀態變數的符號建立理論基礎，但值得注意的是，它們的符號至少是可信的。MP 的正向符號反映出保險實際系統性生產風險的價值。同樣地，UPR 也有正的風險溢價，因為個人希望對沖不確定性增加所導致的總風險溢價的意外增加。由於通貨膨脹的變化具有在投資者之間轉移財富的一般效果，因此並沒有強烈的先驗推定會使 UI 或 DEI 的風險溢價呈現符號，但這些變數的溢價呈現負號，可能意味著股票市場資產被普遍視為對沖其他資產的不利影響，而這些資產的名義價值可能相對較為固定。
As for UTS, the negative risk premium indicates that stocks whose returns are inversely related to increases in long rates oyer short rates are, ceteris paribus, more valuable. One interpretation of this result is that UTS measures a change in the long-term real rate of interest (remember that inflation effects are included in the other variables). After long-term real rates decrease, there is subsequently a lower real return on any form of capital. Investors who want protection against this possibility will place a relatively higher value on assets whose price increases when long-term real rates decline, and such assets will carry a negative risk premium. Thus, stocks whose returns are cor-
就 UTS 而言，負風險溢價顯示，收益與長期利率上升成反比的股票比短 期利率上升的股票更有價值。對於這個結果的一種解釋是，UTS 衡量的是長期實際利率的變化（請記住，通貨膨脹的影響已包含在其他變數中）。長期實際利率降低之後，任何形式的資本的實際回報都會隨之降低。投資人若想要避免這種可能性，就會相對地更重視長期實際利率下降時價格會上升的資產，而這類資產會有負風險溢價。因此，回報與利率相關的股票

TABLE 4 Economic Variables and Pricing (Percent per Month $\times 10$$\times 10$xx10\times 10 ), Multivariate Approach
表 4 經濟變數與定價（每月百分比 $\times 10$$\times 10$xx10\times 10 ），多元法

A