# Structural Shocks, Business Condition Expectations, and Expected Stock Market Returns

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Methods

#### 2.1. Models

#### 2.2. Methods

Notation | Description | Related Data | Data Interval |
---|---|---|---|

$E{R}_{t}$ | The excess market returns | Quarterly return on the S&P 500 index net of three-month treasury bill rate 3^{,}4^{,}5 | 1960–2021 1960–2019 |

Quarterly return on the NYSE composite index net of three-month treasury bill rate 6 | 1960–2021 1960–2019 | ||

${\gamma}_{t}$$\mathrm{and}{\epsilon}_{t}$ | The structural shocks | $\mathrm{Extracted}\mathrm{from}\mathrm{the}\mathrm{VAR}\mathrm{system}\mathrm{of}\Delta {Y}_{t}$$\mathrm{and}\Delta {P}_{t}$ | 1960–2021 1960–2019 |

$\Delta {Y}_{t}$ | The growth rate of Real Gross Domestic Product | Real Gross Domestic Product7 | 1959–2021 1959–2019 |

$\Delta {P}_{t}$ | The growth rate of the implied Gross Domestic Product price deflator | The implied Gross Domestic Product price deflator8 | 1959–2021 1959–2019 |

$Ee{x}_{t}$$\mathrm{and}Er{e}_{t}$ | Business condition expectation | Composite Leading Indicators9 | 1960–2021 1960–2019 |

OECD, Business Tendency Surveys for Manufacturing: Confidence Indicators10 | 1960–2021 1960–2019 |

## 3. Empirical Results of Models (1) and (2)

#### 3.1. Extraction of the Structural Shocks

#### 3.2. The Empirical Results of Model (1)

#### 3.3. The Empirical Results of Model (2)

#### 3.4. Robustness Tests

## 4. The Predictive Power of Models (1) and (2) and Their Economic Significance

#### 4.1. In-Sample Prediction

^{,}18.

$\mathit{D}\mathit{E}{\mathit{F}}_{\mathit{t}}$ | $\mathit{T}\mathit{E}\mathit{R}{\mathit{M}}_{\mathit{t}}$ | $\mathit{d}{\mathit{e}}_{\mathit{t}}$ | $\mathit{s}\mathit{v}\mathit{a}{\mathit{r}}_{\mathit{t}}$ | $\mathit{n}\mathit{t}\mathit{i}{\mathit{s}}_{\mathit{t}}$ | $\mathit{i}{\mathit{k}}_{\mathit{t}}$ | $\widehat{\mathit{c}\mathit{a}{\mathit{y}}_{\mathit{t}}}$ | |
---|---|---|---|---|---|---|---|

Mean | 1.0069 | 2.0666 | −0.7530 | 0.0066 | 0.0097 | 0.0362 | 0.0001 |

Standard Deviation | 0.4300 | 1.8745 | 0.3120 | 0.0111 | 0.0196 | 0.0032 | 0.0039 |

ADF statistic p-value of ADF test | −4.5444 (0.0250) | −2.5885 (0.01) | −2.4799 (0.0144) | −6.7043 (0.01) | −3.3634 (0.01) | −3.5198 (0.01) | −2.3031 (0.0219) |

Number of samples | 248 | 248 | 248 | 248 | 248 | 248 | 248 |

#### 4.2. Long-Horizon Forecast

#### 4.3. Out-of-Sample Forecast

#### 4.4. Economic Significance

- (a)
- Sharpe ratio: The Sharpe ratio is expressed as the ratio of the return of a portfolio to its standard deviation. The return data of Portfolio 1 and Portfolio 2 and the three control portfolios are shown in Table 21.

- (b)
- Jensen’s alpha: Marquering and Verbeek [38] pointed out that the Sharpe ratio does not properly consider time-varying volatility. When measuring dynamic investment strategies, the use of only the standard deviation of the sample (such as the Sharpe ratio) can easily overestimate the risk of the strategy. Therefore, they used Jensen’s alpha to measure the portfolio.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

Name | Description | Source | |

Section 2 | RGDP | U.S. Bureau of Economic Analysis | |

The implied GDP price deflator | U.S. Bureau of Economic Analysis | ||

The turning point of the NBER business cycle | http://www.nber.org/cycles/cyclesmain.html (accessed on 11 September 2022) | ||

S&P 500 index | Yahoo Finance | ||

The three-month treasury bill rate | Board of Governors of the Federal Reserve System (US) | ||

Composite Leading Indicators | OECD | ||

Section 3 | NYSE composite index | Yahoo Finance | |

${D}_{t}$ | Dividend (D12) | http://www.hec.unil.ch/agoyal/ (accessed on 11 September 2022) | |

Another business condition expectation index | OECD | ||

Section 4 | Earnings (E12) | http://www.hec.unil.ch/agoyal/ (accessed on 11 September 2022) | |

$sva{r}_{t}$ | Stock variance | http://www.hec.unil.ch/agoyal/ (accessed on 11 September 2022) | |

$nti{s}_{t}$ | Net securities issuance | http://www.hec.unil.ch/agoyal/ (accessed on 11 September 2022) | |

Moody’s Aaa corporate bond yield | Moody’s | ||

Moody’s Baa corporate bond yield | Moody’s | ||

$i{k}_{t}$ | Investment to Capital Ratio | http://www.hec.unil.ch/agoyal/ (accessed on 11 September 2022) | |

$\widehat{ca{y}_{t}}$ | https://www.sydneyludvigson.com/publications (accessed on 11 September 2022) |

## Conflicts of Interest

## Notes

1 | According to the documents of the OECD, the composite leading indicators from which the business condition expectations in our paper are derived have several inference series and are formed by professional forecasters who try to use all relevant information. Therefore, they belong to rational expectation. The readers can refer to http://www.oecd.org/std/leading-indicators/41629509.pdf (accessed on 16 November 2022). |

2 | The readers should note that the outcomes from 1960–2019 were only calculated with the data from 1960 to 2019. We can also extract the structural shocks using the data set of 1960–2021 and cut the data of 1960–2019 from it but we think that this method is not suitable. |

3 | The adjusted S&P 500 index is from Yahoo Finance. We calculated the quarterly stock market price through the moving average of monthly prices in one quarter. |

4 | The raw data were seasonally adjusted through http://www.seasonal.website/ (accessed on 11 September 2022). All of the raw data that were not seasonally adjusted were adjusted through this website. |

5 | The three-month treasury bill rate was used as the risk-free interest rate. Board of Governors of the Federal Reserve System (US), 3-Month Treasury Bill: Secondary Market Rate [TB3MS], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/TB3MS (accessed on 11 September 2022). |

6 | The adjusted NYSE composite index is from Yahoo Finance. |

7 | U.S. Bureau of Economic Analysis, Real Gross Domestic Product [GDPC1], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GDPC1 (accessed on 11 September 2022). |

8 | U.S. Bureau of Economic Analysis, Gross Domestic Product: Implicit Price Deflator [GDPDEF], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GDPDEF (accessed on 11 September 2022). |

9 | OECD (2022), composite leading indicator (CLI) (indicator). https://doi.org/10.1787/4a174487-en (accessed on 13 September 2022). The raw data are monthly series. We calculated the quarterly data through the moving average of monthly indicators in one quarter. Then, the quarterly data were seasonally adjusted. |

10 | Organization for Economic Cooperation and Development, Business Tendency Surveys for Manufacturing: Confidence Indicators: Composite Indicators: European Commission and National Indicators for the United States [BSCICP02USQ460S], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/BSCICP02USQ460S (accessed on 13 September 2022). |

11 | Our independent variables are stationary. We applied the method of IARM to correct the small-sample bias. Because the whole model of IARM is sophisticated, in this paper, we only provide some simplified procedures to explain the rationale of IARM. The readers can refer to Kim [33]. The author of this paper also provides an R package of IARM. |

12 | The dividend data (D12) are from http://www.hec.unil.ch/agoyal/ (accessed on 11 September 2022). |

13 | The structural shocks were obtained through the vars package of R software. The specific steps can be found in https://rstudio-pubs-static.s3.amazonaws.com/270271_9fbb9b0f8f0c41e6b7e06b0dc2b13b62.html (accessed on 11 September 2022). |

14 | https://cran.r-project.org/web/packages/strucchange/vignettes/strucchange-intro.pdf (accessed on 11 September 2022). For a detailed description of the test method, see the CUSUM process on page 4. |

15 | The vertical lines in Figure 5 and Figure 6 represent the trough dates of the NBER business cycle turning points (http://www.nber.org/cycles/cyclesmain.html) (accessed on 11 September 2022). They include May 1954 (II), April 1958 (II), February 1961 (I), November 1970 (IV), March 1975 (I), July 1980 (III), November 1982 (IV), March 1991 (I), November 2001 (IV), June 2009 (II), and April 2020 (II). |

16 | Federal Reserve Bank of St. Louis, Moody’s Seasoned Aaa Corporate Bond Minus Federal Funds Rate [AAAFFM], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/AAAFFM (accessed on 14 September 2022). |

17 | Moody’s, Moody’s Seasoned Aaa Corporate Bond Yield [AAA], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/AAA (accessed on 13 September 2022). |

18 | Moody’s, Moody’s Seasoned Baa Corporate Bond Yield© [BAA], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/BAA (accessed on 13 September 2022). |

19 | https://www.sydneyludvigson.com/publications (accessed on 11 September 2022). |

20 | http://www.hec.unil.ch/agoyal/ (accessed on 11 September 2022). |

21 | When the ratio is less than or equal to 0, put all funds into risk-free assets; when it is between 0 and 1, put the share equal to the ratio in the stock market index; when the ratio is greater than 1, put all funds into the stock market index. Here, we use the historical variance of the risky assets (e.g., 1960–1992 for the S&P 500) to replace the conditional variance of Marquering and Verbeek [38] in Equation (4). |

22 | Here, the risk-free asset is the three-month treasury bill. |

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Notation | Description | Method of Calculation |
---|---|---|

RGDP | Real Gross Domestic Product | |

$\Delta {Y}_{t}$ | The growth rate of the Real Gross Domestic Product | $\Delta {Y}_{t}=ln\left(RGD{P}_{t}/RGD{P}_{t-1}\right)$ |

GDP | Gross Domestic Product | |

${P}_{t}$ | The implied GDP price deflator | |

$\Delta {P}_{t}$ | The growth rate of the implied GDP price deflator | $\Delta {P}_{t}=ln\left({P}_{t}/{P}_{t-1}\right)$ |

${\gamma}_{t}$ | The permanent shock | Extracted from Vector Autoregressive (VAR) system of $\Delta {Y}_{t}$$\mathrm{and}\Delta {P}_{t}$ |

${\epsilon}_{t}$ | The temporary shock | Extracted from the VAR system of $\Delta {Y}_{t}$$\mathrm{and}\Delta {P}_{t}$ |

$as{p}_{t}$ | The seasonally adjusted stock price | |

$E{R}_{t}$ | The excess stock market return | The stock market return $(ln\left(as{p}_{t}/as{p}_{t-1}\right)$) minus the risk-free interest rate |

$e{x}_{t}$ | A dummy for expansion expectation at time t | The growth rate of one business condition indicator is calculated. When the ratio is non-negative, $e{x}_{t}=1$ |

$r{e}_{t}$ | A dummy for recession expectation at t | The growth rate of one business condition indicator is calculated. When the ratio is negative, $r{e}_{t}=1$ |

$Ee{x}_{t}$ | Expansion expectation | In consideration of the data availability,
$e{x}_{t}$ lagged by two periods is used as the expansion expectation at t $Ee{x}_{t}=E(e{x}_{t}=1|t-2)$ |

$Er{e}_{t}$ | Recession expectation | In consideration of the data availability, $r{e}_{t}$ lagged by two periods is used as the recession expectation at t. $Er{e}_{t}=E(r{e}_{t}=1|t-2)$ |

$Ee{x}_{t}\ast {\gamma}_{t}$ | The permanent shock under the expansion expectation | $Ee{x}_{t}\ast {\gamma}_{t}=E\left(e{x}_{t}=1|t-2\right)\ast {\gamma}_{t}$ |

$Er{e}_{t}\ast {\gamma}_{t}$ | The permanent shock under the recession expectation | $Er{e}_{t}\ast {\gamma}_{t}=E\left(r{e}_{t}=1|t-2\right)\ast {\gamma}_{t}$ |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | The temporary shock under the expansion expectation | $Ee{x}_{t}\ast {\epsilon}_{t}=E\left(e{x}_{t}=1|t-2\right)\ast {\epsilon}_{t}$ |

$Er{e}_{t}\ast {\epsilon}_{t}$ | The temporary shock under the recession expectation | $Er{e}_{t}\ast {\epsilon}_{t}$$=E\left(r{e}_{t}=1|t-2\right)\ast {\epsilon}_{t}$ |

${\mu}_{t+1}$ | The residual term |

ΔY_{t}(1959–2021) | ΔP_{t}(1959–2021) | |

Mean | 0.0073 | 0.0081 |

Standard Deviation | (0.0111) | (0.0058) |

ADF statistic p-value of ADF test | −6.6781 (0.01) | −2.5987 (0.0954) |

Number of samples | 251 | 251 |

$\mathit{E}{\mathit{R}}_{\mathit{t}}$ of S&P 500 (1960–2021) | |
---|---|

Mean | 0.0066 |

Standard Deviation | 0.0654 |

ADF statistic p-value of ADF test | −9.6130 (0.01) |

Number of samples | 248 |

$\mathit{E}{\mathit{R}}_{\mathit{t}+1}$$\mathbf{of}\mathbf{S}\mathbf{P}500$ | |
---|---|

$\mathsf{\delta}$ | 0.0069 (0.0048) |

${\gamma}_{t}$ | 0.0127 * (0.0058) |

${\epsilon}_{t}$ | 0.0042 (0.0066) |

Adjusted R-squared F test p-value of F test | 0.0333 5.237 ** (0.0059) |

$\mathbf{S}\&\mathbf{P}500\phantom{\rule{0ex}{0ex}}\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ | $\mathbf{S}\&\mathbf{P}500\phantom{\rule{0ex}{0ex}}\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ | $\mathbf{S}\&\mathbf{P}500\phantom{\rule{0ex}{0ex}}\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ | |
---|---|---|---|

$\mathsf{\delta}$ | 0.0074 (0.0047) | 0.0067 (0.0051) | 0.0078 ^{†}(0.0046) |

$Ee{x}_{t}\ast {\gamma}_{t}$ | 0.0050 (0.0073) | 0.0051 (0.0072) | |

$Er{e}_{t}\ast {\gamma}_{t}$ | 0.0183 * (0.0087) | 0.0183 * (0.0087) | |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | 0.0045 (0.0081) | 0.0046 (0.0080) | |

$Er{e}_{t}\ast {\epsilon}_{t}$ | 0.0005 (0.0098) | 0.0005 (0.0098) | |

Adjusted R-squared | 0.0351 | −0.0063 | 0.0412 |

F test | 3.218 * | 0.238 | 6.243 ** |

p-value of F test | (0.0135) | (0.7884) | (0.0023) |

^{†}Denotes statistical significance at the 10% level.

NYSE Composite Index $\mathit{E}{\mathit{R}}_{\mathit{t}}$ (1966–2021) | Campbell and Shiller [34] $\mathit{E}{\mathit{R}}_{\mathit{t}}$ (1960–2021) | |
---|---|---|

Mean | 0.0041 | 0.0347 |

Standard Deviation | 0.0687 | 0.0641 |

ADF statistic p-value of ADF test | −9.7177 (0.01) | −7.5867 (0.01) |

Number of samples | 224 | 248 |

S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ IARM Outcomes | NYSE Composite Index $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ IARM Outcomes | Campbell and Shiller [34] $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ IARM Outcomes | |
---|---|---|---|---|

$\mathsf{\delta}$ | 0.0069 (0.0048) | 0.0069 (1.6989) | 0.0049 (1.0982) | 0.0349 *** (8.7506) |

${\gamma}_{t}$ | 0.0127 * (0.0058) | 0.0126 ** (3.0527) | 0.0159 *** (3.4601) | 0.0127 ** (3.1397) |

${\epsilon}_{t}$ | 0.0042 (0.0066) | 0.0039 (0.9645) | 0.0059 (1.3454) | 0.0042 (1.0579) |

Adjusted R-squared F test p-value of F test | 0.0333 5.237 ** (0.0059) | 0.0332 5.2341 ** (0.0059) | 0.0522 7.1158 ** (0.0010) | 0.0353 5.5035 ** (0.0046) |

S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ IARM Outcomes | NYSE Composite Index $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ IARM Outcomes | Campbell and Shiller [34] $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ IARM Outcomes | Another Business Condition Expectation Indicator $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ OLS Outcomes | |
---|---|---|---|---|---|

$\mathsf{\delta}$ | 0.0074 (0.0047) | 0.0077 ^{†}(1.7589) | 0.0064 (1.3701) | 0.0076 ^{†}(1.7589) | 0.0073 (0.0044) |

$Ee{x}_{t}\ast {\gamma}_{t}$ | 0.0050 (0.0073) | 0.0010 (0.1194) | −0.0001 (−0.0102) | 0.0010 (0.1194) | 0.0030 (0.0045) |

$Er{e}_{t}\ast {\gamma}_{t}$ | 0.0183 * (0.0087) | 0.0187 *** (3.4063) | 0.0214 *** (3.6838) | 0.0187 *** (3.4063) | 0.0247 *** (0.0073) |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | 0.0045 (0.0081) | 0.0010 (0.1176) | −0.0014 (−0.1559) | 0.0010 (0.1176) | −0.0018 (0.0062) |

$Er{e}_{t}\ast {\epsilon}_{t}$ | 0.0005 (0.0098) | 0.0021 (0.3906) | 0.0054 (0.9728) | 0.0021 (0.3906) | 0.0122 (0.0075) |

Adjusted R-squared | 0.0351 | 0.0372 | 0.0580 | 0.0372 | 0.0630 |

F test | 3.2180 * | 3.3496 * | 4.4403 ** | 3.3495 * | 5.099 *** |

p-value of F test | (0.0135) | (0.0109) | (0.0018) | (0.0109) | (0.0006) |

^{†}Denotes statistical significance at the 10% level. The number in brackets under the IARM result is the t value.

S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1960–1990) OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1960–1990) IARM Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1991–2021) OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1991–2021) IARM Outcomes | |
---|---|---|---|---|

$\mathsf{\delta}$ | −0.0034 (0.0063) | −0.0022 (−0.4144) | 0.0175 ** (0.0047) | 0.0175 ** (3.2609) |

${\gamma}_{t}$ | 0.0139 * (0.0065) | 0.0108 ^{†}(1.7130) | 0.0093 (0.0101) | 0.0092 (1.5670) |

${\epsilon}_{t}$ | 0.0055 (0.0065) | −0.0071 (−1.0654) | 0.0145 ^{†}(0.0080) | 0.0146 ** (2.6005) |

Adjusted R-squared F test p-value of F test | 0.0350 3.1550 * (0.0463) | 0.0329 3.0730 * (0.0500) | 0.0862 6.7540 ** (0.0017) | 0.0884 6.9162 ** (0.0014) |

^{†}Denotes statistical significance at the 10% level. The number in brackets under the IARM result is the t value.

S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1960–1990) OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1960–1990) IARM Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1991–2021) OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1991–2021) IARM Outcomes | |
---|---|---|---|---|

$\mathsf{\delta}$ | −0.0023 (0.0048) | −0.0009 (−0.3975) | 0.0174 *** (0.0049) | 0.0177 ** (3.1785) |

$Ee{x}_{t}\ast {\gamma}_{t}$ | 0.0017 (0.0083) | −0.0054 (−0.5365) | 0.0283 * (0.0134) | 0.0224 (1.5655) |

$Er{e}_{t}\ast {\gamma}_{t}$ | 0.0232 *** (0.0055) | 0.0236 ** (2.7640) | 0.0011 (0.0123) | 0.0038 (0.4673) |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | 0.0030 (0.0087) | −0.0055 (−0.4868) | 0.0197 ^{†}(0.0105) | 0.0225 ^{†}(1.8722) |

$Er{e}_{t}\ast {\epsilon}_{t}$ | −0.0128 (0.0092) | −0.0097 (−1.0943) | 0.0201 ^{†}(0.0106) | 0.0184 * (2.3716) |

Adjusted R-squared | 0.0690 | 0.0606 | 0.0904 | 0.0843 |

F test | 3.222 * | 2.9182 * | 3.981 ** | 3.7611 ** |

p-value of F test | (0.0151) | (0.0244) | (0.0046) | (0.0066) |

^{†}Denotes statistical significance at the 10% level. The number in brackets under the IARM result is the t value.

S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1960–1990) OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1960–1990) IARM Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1991–2019) OLS Outcomes | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$(1991–2019) IARM Outcomes | |
---|---|---|---|---|

$\mathsf{\delta}$ | −0.0030 (0.0044) | −0.0035 (−0.6315) | 0.0183 *** (0.0053) | 0.0181 *** 3.6853) |

$Ee{x}_{t}\ast {\gamma}_{t}$ | 0.0037 (0.0054) | 0.0021 (0.2795) | 0.0150 ^{†}(0.0087) | 0.0127 (1.3116) |

$Er{e}_{t}\ast {\gamma}_{t}$ | 0.0239 *** (0.0069) | 0.0229 * (3.1432) | 0.0455 * (0.0127) | 0.0369 ** (3.0898) |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | 0.0025 (0.0088) | 0.0003 (0.0316) | 0.0106 (0.0087) | 0.0128 1.2401) |

$Er{e}_{t}\ast {\epsilon}_{t}$ | −0.0067 (0.0079) | −0.0032 (−0.4981) | 0.0508 *** (0.0099) | 0.0509 *** (5.7070) |

Adjusted R-squared | 0.0621 | 0.0477 | 0.3051 | 0.2619 |

F test | 2.9710 * | 2.4895 * | 13.3 *** | 10.9363 *** |

p-value of F test | (0.0224) | (0.0473) | (7.6610 × 10^{−9}) | (1.9310 × 10^{−7}) |

^{†}Denotes statistical significance at the 10% level. The number in brackets under the IARM result is the t value.

S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ | |
---|---|

$\mathsf{\delta}$ | 0.1310 * (0.0650) |

${\gamma}_{t}$ | 0.0117 * (0.0046) |

${\epsilon}_{t}$ | 0.0040 (0.0059) |

$d{e}_{t}$ | −0.0056 (0.0146) |

$sva{r}_{t}$ | −0.2119 (0.7530) |

$nti{s}_{t}$ | −0.2850 (0.2351) |

$DE{F}_{t}$ | 0.0005 (0.0096) |

$TER{M}_{t}$ | 0.0028 (0.0028) |

$i{k}_{t}$ | −3.5980 * (1.6349) |

$\widehat{ca{y}_{t}}$ | −0.0661 (0.1264) |

Adjusted R-squared | 0.0662 |

F test p-value of F test | 2.938 ** (0.0025) |

$\mathbf{S}\&\mathbf{P}500\phantom{\rule{0ex}{0ex}}\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ | |
---|---|

$\mathsf{\delta}$ | 0.1435 (0.0648) |

$Ee{x}_{t}\ast {\gamma}_{t}$ | 0.0069 (0.0069) |

$Er{e}_{t}\ast {\gamma}_{t}$ | 0.01688 * (0.0075) |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | 0.0071 (0.0067) |

$Er{e}_{t}\ast {\epsilon}_{t}$ | −0.0003 (0.0089) |

$d{e}_{t}$ | −0.0052 (0.0152) |

$sva{r}_{t}$ | −0.1955 (0.7704) |

$nti{s}_{t}$ | −0.3077 (0.2353) |

$DE{F}_{t}$ | −0.0006 (0.0093) |

$TER{M}_{t}$ | 0.0021 (0.0026) |

$i{k}_{t}$ | −3.8560 * (1.6638) |

$\widehat{ca{y}_{t}}$ | −0.0631 (0.1204) |

Adjusted R-squared | 0.0665 |

F test p-value of F test | 2.58 ** (0.0042) |

$\mathit{L}{\mathit{R}}_{\mathit{t}+\mathit{i}}\mathbf{of}\mathbf{S}\mathbf{P}500$ | |||
---|---|---|---|

$i=12$ | $i=14$ | $i=16$ | |

$\mathsf{\delta}$ | 0.0680 * (0.0278) (0.0375, 0.0408) | 0.0781 ** (0.0178) (0.0163, 0.0197) | 0.0878 ** (0.0192) (0.0177, 0.0209) |

${\gamma}_{t}$ | −0.0225 (0.0187) (0.0136, 0.0254) | −0.0219 (0.0185) (0.0121, 0.0240) | −0.0263 (0.0203) (0.0142, 0.0263) |

${\epsilon}_{t}$ | −0.0554 ** (0.0199) (0.0159, 0.0277) | −0.0649 ** (0.0186) (0.0121, 0.0240) | −0.0768 ** (0.0200) (0.0138, 0.0257) |

Adjusted R-squared chi-squared test p-value of chi-squared test | 0.0201 6.4 * (0.042) | 0.0254 13.0 ** (0.0015) | 0.0327 18.1 *** (0.0001) |

$\mathit{L}{\mathit{R}}_{\mathit{t}+\mathit{i}}\mathbf{of}\mathbf{S}\mathbf{P}500$ | |||
---|---|---|---|

$i=12$ | $i=14$ | $i=16$ | |

$\mathsf{\delta}$ | 0.0715 ** (0.0170) (0.0156, 0.0190) | 0.0797 ** (0.0183) (0.0170, 0.0204) | 0.0889 ** (0.0197) (0.0187, 0.0220) |

$Ee{x}_{t}\ast {\gamma}_{t}$ | −0.0594 ^{†}(0.0354) (0.0350, 0.0553) | −0.0470 (0.0337) (0.0283, 0.0509) | −0.0515 (0.0334) (0.0262, 0.0492) |

$Er{e}_{t}\ast {\gamma}_{t}$ | 0.0014 (0.0188) (0.0008, 0.0228) | −0.0017 (0.0201) (0.0015, 0.0237) | −0.0043 (0.0259) (0.0115, 0.0343) |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | −0.0896 * (0.0372) (0.0375, 0.0590) | −0.0759 ** (0.0356) (0.0304, 0.0545) | −0.0837 * (0.0362) (0.0307, 0.0542) |

$Er{e}_{t}\ast {\epsilon}_{t}$ | −0.0368 ^{†}(0.0204) (0.0024, 0.0255) | −0.0616 *** (0.0215) (0.0027, 0.0265) | −0.0761 ** (0.0253) (0.0095, 0.0327) |

Adjusted R-squared chi-squared test p-value of chi-squared test | 0.0221 9.8 * (0.0430) | 0.0202 26.6 *** (2.4 × 10 ^{−5}) | 0.0270 25.4 *** (4.1 × 10 ^{−5}) |

^{†}Denotes statistical significance at the 10% level.

$\mathit{L}{\mathit{R}}_{\mathit{t}+\mathit{i}}\mathbf{of}\mathbf{S}\mathbf{P}500$ | ||||
---|---|---|---|---|

$\mathit{i}=4$ | $\mathit{i}=6$ | $\mathit{i}=14$ | $\mathit{i}=16$ | |

$\mathsf{\delta}$ | 0.0210 * (0.0099) (0.0087, 0.0112) | 0.0101 (0.0136) (0.0110, 0.0139) | 0.0681 * (0.0311) (0.0424, 0.0460) | 0.0750 * (0.0332) (0.0455, 0.0491) |

${\gamma}_{t}$ | 0.0227 * (0.0098) (0.0067, 0.0118) | 0.0331 ^{†}(0.0171) (0.0097, 0.0160) | 0.0099 (0.0140) (0.0048, 0.0131) | 0.0145 (0.0155) (0.0072, 0.0152) |

${\epsilon}_{t}$ | −0.0130 (0.0104) (0.0078, 0.0132) | −0.0326 * (0.0138) (0.0093, 0.0159) | −0.0300 * (0.0134) (0.0036, 0.0123) | −0.0393 ** (0.0147) (0.0049, 0.0137) |

Adjusted R-squared chi-squared test p-value of chi-squared test | 0.0198 5.9 ^{†}(0.051) | 0.0241 5.6 ^{†}(0.061) | 0.0039 13 ** (0.0015) | 0.0106 18.1 ** (0.0079) |

^{†}Denotes statistical significance at the 10% level.

$\mathit{L}{\mathit{R}}_{\mathit{t}+\mathit{i}}\mathbf{of}\mathbf{S}\mathbf{P}500$ | |||
---|---|---|---|

$\mathit{i}=4$ | $\mathit{i}=14$ | $\mathit{i}=16$ | |

$\mathsf{\delta}$ | 0.0197 (0.0160) (0.0207, 0.0231) | 0.0670 ** (0.0186) (0.0172, 0.0208) | 0.0729 ** (0.0197) (0.0181, 0.0220) |

$Ee{x}_{t}\ast {\gamma}_{t}$ | 0.0133 (0.0129) (0.0056, 0.0158) | −0.0068 (0.0282) (0.0208, 0.0374) | −0.0087 (0.0276) (0.0180, 0.0349) |

$Er{e}_{t}\ast {\gamma}_{t}$ | 0.0364 * (0.0174) (0.0140, 0.0249) | 0.0296 (0.0181) (0.0001, 0.0174) | 0.0437 ^{†}(0.0228) (0.0073, 0.0257) |

$Ee{x}_{t}\ast {\epsilon}_{t}$ | −0.0041 (0.0194) (0.0184, 0.0288) | −0.0261 (0.0302) (0.0252, 0.0415) | −0.0289 (0.0304) (0.0227, 0.0404) |

$Er{e}_{t}\ast {\epsilon}_{t}$ | −0.0185 (0.0113) (0.0029, 0.0130) | −0.0338 * (0.0171) (−0.0022, 0.0155) | −0.0485 * (0.0209) (0.0038, 0.0213) |

Adjusted R-squared chi-squared test p-value of chi-squared test | 0.0184 8.1 ^{†}(0.087) | −0.0049 10.5 * (0.033) | 0.0066 12.1 * (0.017) |

^{†}Denotes statistical significance at the 10% level.

S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1993–2021) | NYSE Composite Index $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1991–2021) | S&P 500 $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1993–2019) | NYSE Composite Index $\mathit{E}{\mathit{R}}_{\mathit{t}+1}$ (1991–2019) | |
---|---|---|---|---|

Model (1) | Model (1) | Model (1) | Model (1) | |

Root mean square error Out-of-sample R-squared | 0.0652 −0.0581 | 0.0657 −0.0520 | 0.0644 −0.0427 | 0.0640 −0.0219 |

Model (2) | Model (2) | Model (2) | Model (2) | |

Root mean square error Out-of-sample R-squared | 0.0668 −0.1133 | 0.0661 −0.0673 | 0.0642 −0.0363 | 0.0638 −0.0168 |

The control model | The control model | The control model | The control model | |

Root mean square error | 0.0651 | 0.0653 | 0.0643 | 0.0643 |

Mean | Standard Deviation | Sharpe Ratio | |
---|---|---|---|

S&P 500 Index (1993–2021) | |||

Portfolio 1 | 0.0094 | 0.0137 | 0.6826 |

Portfolio 2 | 0.0076 | 0.0213 | 0.3569 |

Control portfolio 1 | 0.0205 | 0.0636 | 0.3229 |

Control portfolio 2 | 0.0131 | 0.0320 | 0.4087 |

Control portfolio 3 | 0.0056 | 0.0052 | 0 |

NYSE Composite Index (1991–2021) | |||

Portfolio 1 | 0.0075 | 0.0107 | 0.7015 |

Portfolio 2 | 0.0076 | 0.0152 | 0.5013 |

Control portfolio 1 | 0.0165 | 0.0646 | 0.2558 |

Control portfolio 2 | 0.0111 | 0.0327 | 0.3413 |

Control portfolio 3 | 0.0058 | 0.0051 | 0 |

Mean | Standard Deviation | Sharpe Ratio | |
---|---|---|---|

S&P 500 Index (1993–2019) | |||

Portfolio 1 | 0.0095 | 0.0132 | 0.7207 |

Portfolio 2 | 0.0129 | 0.0211 | 0.6135 |

Control portfolio 1 | 0.0184 | 0.06364 | 0.2895 |

Control portfolio 2 | 0.0122 | 0.0321 | 0.3800 |

Control portfolio 3 | 0.0060 | 0.0051 | 0 |

NYSE Composite Index (1991–2019) | |||

Portfolio 1 | 0.0085 | 0.0120 | 0.7038 |

Portfolio 2 | 0.0082 | 0.0153 | 0.5345 |

Control portfolio 1 | 0.0158 | 0.0640 | 0.2463 |

Control portfolio 2 | 0.0110 | 0.0325 | 0.3375 |

Control portfolio 3 | 0.0061 | 0.0051 | 0 |

${\mathit{\alpha}}_{\mathit{p}}$ Model (3) | ${\mathit{\phi}}_{1}$ Model (4) | |
---|---|---|

S&P 500 index (1993–2021) | ||

Portfolio 1 | 0.0082 *** (6.1202) | 0.2952 (1.6507) |

Portfolio 2 | 0.0049 * (2.3546) | −0.0569 (−0.2706) |

NYSE Composite Index (1991–2021) | ||

Portfolio 1 | 0.0068 *** (5.1115) | 0.0282 (0.4270) |

Portfolio 2 | 0.0064 *** (4.6619) | 0.2598 * (2.4505) |

${\mathit{\alpha}}_{\mathit{p}}$ Model (3) | ${\mathit{\phi}}_{1}$ Model (4) | |
---|---|---|

S&P 500 index (1993–2019) | ||

Portfolio 1 | 0.0085 *** (5.8560) | 0.3671 ** (3.0519) |

Portfolio 2 | 0.0082 *** (5.8761) | 0.4967 ^{†}(1.9693) |

NYSE Composite Index (1991–2019) | ||

Portfolio 1 | 0.0076 *** (5.5115) | 0.2266 ** (3.0443) |

Portfolio 2 | 0.0072 *** (5.3250) | 0.4085 ^{†}(1.7516) |

^{†}Denotes statistical significance at the 10% level.

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## Share and Cite

**MDPI and ACS Style**

Chen, W.; Liu, M.
Structural Shocks, Business Condition Expectations, and Expected Stock Market Returns. *Systems* **2022**, *10*, 228.
https://doi.org/10.3390/systems10060228

**AMA Style**

Chen W, Liu M.
Structural Shocks, Business Condition Expectations, and Expected Stock Market Returns. *Systems*. 2022; 10(6):228.
https://doi.org/10.3390/systems10060228

**Chicago/Turabian Style**

Chen, Weizhong, and Mingming Liu.
2022. "Structural Shocks, Business Condition Expectations, and Expected Stock Market Returns" *Systems* 10, no. 6: 228.
https://doi.org/10.3390/systems10060228