# Modelling and Forecasting Stock Price Movements with Serially Dependent Determinants

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## Abstract

**:**

## 1. Introduction

## 2. A Review of the Ordered Probit Model

#### Models for Correlated Errors and Explanatory Variables

## 3. Data, Variables and ACD Model

#### 3.1. Data Description and ACD Model

#### 3.2. Sample Statistics

## 4. Empirical Evidence

#### 4.1. Ordered Probit Model Estimation

#### 4.2. Price Impact of a Trade

#### 4.3. Diagnostics

## 5. Forecasting the Direction of Price Change

#### 5.1. Out-of-Sample Multi-Step Ahead Forecasts with Disaggregated Predictions of Individual Explanatory Variables

#### 5.2. Forecast Performance of the Ordered Probit Model

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Models for Errors and Explanatory Variables

#### Appendix A.1. ACD Specification to Model Transaction Durations

#### Appendix A.2. GARCH Specification to Model Heteroscedasticity

#### Appendix A.3. ARIMA Model

#### Appendix A.4. Long Memory ARFIMA Model

#### Appendix A.5. Multinomial Logistic Regression

## References

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**Figure 4.**Distribution of the number of trades over the three categories of price change in terms of ticks, for all stocks during the period.

**Figure 5.**Distribution of estimated probabilities of direction of price change conditioned on constant, increasing and decreasing past price changes.

**Figure 6.**Distribution of estimated probabilities of direction of price change conditioned on constant, increasing and decreasing past price changes.

**Figure 7.**In-sample estimated probabilities of direction of price change for 100 observations of CBA.

**Figure 8.**Out of-sample estimated probabilities of direction of price change for 100 observations of CBA, based on actual regressor values.

**Table 1.**Descriptive statistics of the variables considered in the ordered probit model for all the stocks, for the period from 16 January 2014 to 15 April 2014.

Statistic | AGL | BHP | CBA | NCP | TLS | WES | WPL |
---|---|---|---|---|---|---|---|

Price (AUD) | |||||||

Max price | 16.15 | 39.79 | 77.87 | 20.17 | 5.29 | 43.93 | 39.5 |

Min price | 14.71 | 35.06 | 72.15 | 16.92 | 4.96 | 40.88 | 36.54 |

Price Change (%) | |||||||

Mean | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Std.dev | 0.0376 | 0.0197 | 0.0124 | 0.0728 | 0.0765 | 0.0181 | 0.0183 |

Duration (Seconds) | |||||||

Mean | 9.59 | 3.51 | 3.49 | 24.76 | 8.04 | 4.26 | 5.30 |

Std.dev | 19.01 | 7.37 | 7.56 | 49.91 | 12.47 | 9.08 | 11.08 |

Trade Volume | |||||||

Mean | 395 | 710 | 285 | 161 | 6983 | 281 | 318 |

Std.dev | 2206 | 3622 | 2183 | 711 | 39,379 | 1370 | 1290 |

Shares at the Best Bid Price | |||||||

Mean | 4451 | 5498 | 1579 | 877 | 941,002 | 1841 | 1983 |

Std.dev | 5027 | 6464 | 2899 | 1994 | 603,015 | 2504 | 2406 |

Shares at the Best Ask Price | |||||||

Mean | 4399 | 5513 | 1808 | 977 | 992,906 | 1945 | 2100 |

Std.dev | 5409 | 7544 | 5053 | 1649 | 642,204 | 2775 | 2719 |

Market Index Returns, ASX200 | |||||||

Mean | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Std.dev | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

Trade Imbalance | |||||||

Mean | −0.0268 | −0.0119 | 0.0094 | −0.0623 | 0.0094 | 0.0224 | 0.0004 |

Std.dev | 0.4653 | 0.4590 | 0.4401 | 0.4863 | 0.5082 | 0.4564 | 0.4446 |

Trade Direction (%) | |||||||

Buyer initiated | 40.9 | 41.0 | 44.7 | 43.9 | 27.0 | 44.2 | 44.6 |

Seller initiated | 41.6 | 41.0 | 42.2 | 48.1 | 27.6 | 40.6 | 41.9 |

**Table 2.**The coefficient estimates of an ACD (1,1) model with Standardised Weibull errors fitted for the stocks. The conditional expected duration where ${x}_{k}$ is the adjusted duration. $\alpha $ is the shape parameter of the Weibull distribution.

Parameter | AGL | BHP | CBA | NCP | TLS | WES | WPL |
---|---|---|---|---|---|---|---|

${\alpha}_{0}$ | 0.3177 | 0.0024 | 0.3178 | 0.0348 | 0.0291 | 0.3349 | 0.0030 |

(21.92 *) | (10.16 *) | (32.47 *) | (10.03 *) | (28.82 *) | (23.33 *) | (6.65 *) | |

${\alpha}_{1}$ | 0.3113 | 0.0110 | 0.2195 | 0.1476 | 0.2180 | 0.1652 | 0.0201 |

(27.91 *) | (20.61 *) | (41.03 *) | (15.53 *) | (59.84 *) | (29.71 *) | (18.48 *) | |

$\beta $ | 0.4764 | 0.9865 | 0.4949 | 0.8524 | 0.7820 | 0.5220 | 0.9785 |

(26.80 *) | (1371.66 *) | (40.08 *) | (89.69 *) | (214.64 *) | (30.26 *) | (747.16 *) | |

$\alpha $ | 0.2523 | 0.4295 | 0.4258 | 0.4369 | 0.5756 | 0.4194 | 0.4046 |

(427.88 *) | (726.73 *) | (731.49 *) | (255.94 *) | (476.06 *) | (672.39 *) | (582.99 *) |

**Table 3.**Coefficient estimates ${\beta}_{i}$, of ordered probit model on direction of price change based on 12 explanatory variables for the selected stocks. The sampling period was 16 January 2014 to 15 January 2014. Z statistics are given within parentheses for each parameter.

Parameter | AGL | BHP | CBA | NCP | TLS | WES | WPL |
---|---|---|---|---|---|---|---|

Obs. | 114,318 | 316,547 | 317,761 | 41,085 | 137,323 | 260,954 | 205,651 |

${Y}_{k-1}$ | −2.2281 | −1.0240 | −0.7915 | −0.7637 | −1.7745 | −1.4750 | −1.6261 |

(−72.21 *) | (−130.16 *) | (−119.64 *) | (−54.32 *) | (−53.16 *) | (−143.59 *) | (−113.69 *) | |

${Y}_{k-2}$ | −1.1262 | −0.3614 | −0.3247 | −0.2554 | −0.9070 | −0.7413 | −0.8578 |

(−37.38 *) | (−50.67 *) | (−52.35 *) | (−18.92 *) | (−21.15 *) | (−72.42 *) | (−62.50 *) | |

${Y}_{k-3}$ | −0.5905 | −0.1142 | −0.1153 | −0.1405 | −0.4252 | −0.3533 | −0.4137 |

(−19.79 *) | (−16.65 *) | (−19.56 *) | (−10.50 *) | (−12.21 *) | (−33.69 *) | (−30.20 *) | |

$TI$ | 0.9572 | 1.2730 | −0.2832 | −0.1256 | −1.2319 | 0.8899 | 0.9831 |

(67.69 *) | (285.32*) | (−88.07 *) | (−8.95 *) | (−38.61 *) | (165.00 *) | (146.16 *) | |

$Sprd$ | 0.0479 | 0.0342 | 0.0078 | 0.0117 | 0.0703 | 0.0238 | 0.0261 |

(3.16 *) | (6.71 *) | (2.60 *) | (1.85 **) | (1.21) | (0.95) | (4.83 *) | |

$LVol$ | 0.0047 | 0.0090 | 0.0050 | 0.0167 | 0.0118 | 0.0060 | 0.0043 |

(1.16) | (6.13 *) | (3.67 *) | (4.92 *) | (3.04 *) | (3.03 *) | (1.85 **) | |

$LBAV$ | 0.0801 | 0.1019 | 0.0337 | −0.0312 | −0.0515 | 0.0398 | 0.0607 |

(12.36 *) | (48.39 *) | (20.31 *) | (−6.94 *) | (−4.76 *) | (15.25 *) | (19.05 *) | |

$LBBV$ | −0.0760 | −0.1097 | −0.0397 | 0.0252 | 0.0425 | −0.0575 | −0.0744 |

(−12.03 *) | (−53.04 *) | (−23.81 *) | (5.66 *) | (3.32 *) | (−21.15 *) | (−21.51 *) | |

$TIB$ | 0.0488 | 0.0647 | 0.0929 | 0.0457 | 0.1916 | 0.0986 | 0.0696 |

(2.52 *) | (9.70 *) | (15.89 *) | (3.04 *) | (9.47 *) | (11.50 *) | (6.77 *) | |

$TI\ast \psi $ | −0.2246 | −0.3764 | 0.7304 | 0.0353 | −0.0546 | −0.2506 | −0.2631 |

(−31.79 *) | (−119.95 *) | (203.38 *) | (4.24 *) | (−3.91 *) | (−62.24 *) | (−56.41 *) | |

$TI\ast \u03f5$ | −0.0535 | −0.0608 | 0.0706 | 0.0013 | −0.0213 | −0.0314 | −0.0328 |

(−11.51 *) | (−37.49 *) | (55.74 *) | (0.40) | (−4.01 *) | (−14.58 *) | (−12.73 *) | |

$RIndx$ | 262.0988 | 110.1514 | 194.2312 | 307.72 | −107.5324 | 342.8466 | 139.1455 |

(3.30 *) | (2.47 *) | (3.71 *) | (7.49 *) | (−0.87) | (6.52 *) | (2.72 *) | |

${\alpha}_{1}$ | −2.8628 | −2.1469 | −1.6552 | −1.5723 | −4.5651 | −2.3399 | −2.4639 |

${\alpha}_{2}$ | 2.9999 | 2.1869 | 1.6699 | 1.7676 | 5.1375 | 2.2403 | 2.3768 |

$\mathit{Pseudo}-{\mathit{R}}^{2}$ | 0.3203 | 0.3339 | 0.2226 | 0.2068 | 0.3223 | 0.2589 | 0.2833 |

**Table 4.**Coefficient estimates of GARCH parameters of the conditional variance of the residuals for all stocks. $\omega $, constant; $\kappa $, GARCH parameters; $\delta $, ARCH parameters

Parameter | AGL | BHP | CBA | NCP | TLS | WES | WPL |
---|---|---|---|---|---|---|---|

Obs. | 114,318 | 316,547 | 317,761 | 41,085 | 137,323 | 260,954 | 205,651 |

$\omega $ | 0.4081 | 0.0468 | 0.0022 | 0.0598 | 0.0204 | 0.3242 | 0.3282 |

(0.3419) | (0.5782) | (0.4748) | (0.3415) | (0.1226) | (0.5995) | (0.4998) | |

${\kappa}_{1}$ | 0.3194 | 0.3803 | 0.9723 | 0.8255 | 0.9175 | 0.2445 | 0.2406 |

(0.2182) | (0.3205) | (46.8638) | (2.4207) | (2.9561) | (0.2668) | (0.2209) | |

${\kappa}_{2}$ | 0.5157 | 0.2045 | 0.2172 | ||||

(0.4516) | (0.2272) | (0.2019) | |||||

${\delta}_{1}$ | 0.2523 | 0.0429 | 0.0244 | 0.0873 | 0.0615 | 0.1735 | 0.1660 |

(0.6870) | (0.9682) | (1.4103) | (0.6766) | (0.0.2994) | (1.3347) | (1.1188) |

**Table 5.**Cross-autocorrelation coefficents ${\widehat{v}}_{j},j=1,\phantom{\rule{0.166667em}{0ex}}$…$,12$ of generalised residuals with lagged generalised fitted price changes.

Parameter | AGL | BHP | CBA | NCP | TLS | WES | WPL |
---|---|---|---|---|---|---|---|

${\widehat{v}}_{1}$ | −0.0025 | −0.0002 | −0.0015 | −0.0004 | 0.0004 | −0.0015 | −0.0004 |

${\widehat{v}}_{2}$ | −0.0057 | 0.0012 | −0.0015 | −0.0002 | −0.0012 | −0.0003 | 0.0009 |

${\widehat{v}}_{3}$ | −0.0103 | −0.0005 | −0.0016 | 0.0008 | −0.0008 | 0.0010 | 0.0015 |

${\widehat{v}}_{4}$ | −0.0058 | 6.19 × 10${}^{-6}$ | −0.0018 | −0.0029 | −0.0028 | −0.0004 | 0.0013 |

${\widehat{v}}_{5}$ | −0.0045 | 0.0006 | −0.0018 | −0.0022 | −0.0039 | 0.0005 | −0.0017 |

${\widehat{v}}_{6}$ | −0.0056 | −0.0001 | −0.0020 | −0.0025 | 0.0016 | 0.0031 | 0.0018 |

${\widehat{v}}_{7}$ | 0.0009 | −0.0008 | −0.0018 | 0.0002 | −0.0015 | 0.0034 | 0.0001 |

${\widehat{v}}_{8}$ | 0.0029 | 0.0001 | −0.0017 | 0.0043 | −0.0039 | 0.0010 | 0.0003 |

${\widehat{v}}_{9}$ | 0.0001 | −7.76 × 10${}^{-5}$ | −0.0017 | 0.0057 | −0.0023 | 0.0036 | −0.0023 |

${\widehat{v}}_{10}$ | 0.0047 | 0.0003 | −0.0015 | 0.0039 | −0.0021 | 0.0030 | 0.0042 |

${\widehat{v}}_{11}$ | 0.0076 | 0.0020 | −0.0013 | 0.0025 | −0.0033 | 0.0009 | 0.0017 |

${\widehat{v}}_{12}$ | 0.0011 | 0.0014 | −0.0011 | 0.0014 | −0.0041 | 0.0024 | −0.0002 |

**Table 6.**Coefficient estimates of autoregressive model parameters fitted to selected independent variables. The t statistics are given within parentheses. Illustrative examples include a long memory and a short memory model for $LVol$ and $Sprd$ for the stock AGL. $\mathit{d}$, long memory parameter; $\varphi $, AR parameters; $\theta $, MA parameters.

Parameter | LVol | Spread |
---|---|---|

(ARFIMA) | (ARMA) | |

c | 0.0030 (7.30) | |

d | 0.1867 (68.50) | |

${\varphi}_{1}$ | 0.0082 (15.72) | 1.7555 (160.732) |

${\varphi}_{2}$ | −0.7581 (−71.06) | |

${\theta}_{1}$ | −0.0079 (−35.59) | −1.3455 (−123.55) |

${\theta}_{2}$ | 0.2774 (41.16) | |

${\theta}_{3}$ | 0.0621 (15.35) | |

${\theta}_{4}$ | 0.0136 (3.67) | |

${\theta}_{5}$ | 0.0052 (1.80) |

**Table 7.**Coefficient estimates of multinomial logistic regression model parameters fitted to Trade indicator ($TI$) of AGL. The base category is 1. Z statistics are given in parentheses.

Independent | Category | |
---|---|---|

Variable | −1 | 0 |

c | 0.1449 (2.63) | −1.9155 (−26.87) |

$d{p}_{k-1}$ | 0.2095 (5.08) | 0.1128 (1.98) |

$lbb{v}_{k-1}$ | −0.3146 (−60.57) | −0.0746 (−10.41) |

$lba{v}_{k-1}$ | 0.3001 (60.42) | 0.2302 (35.96) |

$T{I}_{k-1}$ | −0.9664 (−119.26) | −0.4632 (−42.96) |

**Table 8.**In-sample predictions of direction of price change for the last one week period of the training sample from 8 April 2014 to 14 April 2014.

Parameter | AGL | BHP | CBA | NCP | TLS | WES | WPL | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

One Week—08/04–14/04 | |||||||||||||||

$\mathit{Observations}$ | 114,318 | 316,547 | 317,761 | 41,085 | 137,323 | 260,954 | 205,651 | ||||||||

$\mathit{accuracy}(\%)$ | 97.80 | 86.20 | 85.11 | 85.24 | 98.75 | 92.19 | 94.76 | ||||||||

−1 | 40.00 | 23.79 | 36.03 | 25.48 | 39.29 | 27.11 | 38.45 | ||||||||

0 | 99.79 | 98.67 | 98.14 | 98.64 | 99.65 | 99.51 | 99.78 | ||||||||

+1 | 45.40 | 24.29 | 35.73 | 32.01 | 36.00 | 33.57 | 37.09 | ||||||||

Actual | Forecast | No. | % | No. | % | No. | % | No. | % | No. | % | No. | % | No. | % |

−1 | +1 | 0 | 0 | 1 | 0.05 | 6 | 0.32 | 1 | 0.42 | 0 | 0 | 0 | 0 | 0 | 0 |

+1 | −1 | 0 | 0 | 0 | 0 | 3 | 0.17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 9.**Out-of sample forecasts of direction of price change for the last day of the sample, 15 April 2014. First panel contains one-step ahead forecasts based on actual explanatory variables and the second panel, muti-step ahead with predicted variables.

Parameter | AGL | BHP | CBA | NCP | TLS | WES | WPL | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

One-Step Ahead—15/04 | |||||||||||||||

$\mathit{Observations}$ | 1151 | 3319 | 4073 | 417 | 1218 | 2518 | 3638 | ||||||||

$\mathit{accuracy}\phantom{\rule{0.166667em}{0ex}}(\%)$ | 97.83 | 85.90 | 79.65 | 75.30 | 99.67 | 88.28 | 91.70 | ||||||||

−1 | 35.29 | 27.40 | 26.80 | 21.54 | 0.00 | 32.54 | 32.77 | ||||||||

0 | 99.73 | 98.47 | 97.31 | 96.82 | 99.92 | 98.95 | 99.47 | ||||||||

+1 | 38.89 | 23.84 | 26.88 | 37.68 | 0.00 | 38.03 | 35.53 | ||||||||

Actual | Forecast | No. | % | No. | % | No. | % | No. | % | No. | % | No. | % | No. | % |

−1 | +1 | 0 | 0 | 0 | 0 | 6 | 1.2 | 0 | 0 | 0 | 0 | 2 | 1.2 | 0 | 0 |

+1 | −1 | 0 | 0 | 0 | 0 | 1 | 0.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Multi-Step Ahead—15/04 | |||||||||||||||

(a) All transactions | |||||||||||||||

$\mathit{Observations}$ | 1151 | 3319 | 4073 | 417 | 1218 | 2518 | 3638 | ||||||||

$\mathit{accuracy}\phantom{\rule{0.166667em}{0ex}}(\%)$ | 97.74 | 82.74 | 74.93 | 67.87 | 47.46 | 87.41 | 90.07 | ||||||||

(b) 100-step ahead | |||||||||||||||

$\mathit{accuracy}\phantom{\rule{0.166667em}{0ex}}(\%)$ | 97.48 | 83.25 | 80.55 | 74.82 | 79.97 | 89.87 | 92.25 | ||||||||

−1 | 30.77 | 25.34 | 21.94 | 20.00 | 0.00 | 23.78 | 20.43 | ||||||||

0 | 98.93 | 94.94 | 99.87 | 100.00 | 80.16 | 99.14 | 99.88 | ||||||||

+1 | 38.46 | 29.18 | 23.72 | 23.19 | 0.00 | 27.45 | 25.71 | ||||||||

Actual | Forecast | No. | % | No. | % | No. | % | No. | % | No. | % | No. | % | No. | % |

−1 | +1 | 0 | 0 | 2 | 0.68 | 3 | 0.58 | 0 | 0 | 1 | 50 | 0 | 0 | 0 | 0 |

+1 | −1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yatigammana, R.; Peiris, S.; Gerlach, R.; Allen, D.E.
Modelling and Forecasting Stock Price Movements with Serially Dependent Determinants. *Risks* **2018**, *6*, 52.
https://doi.org/10.3390/risks6020052

**AMA Style**

Yatigammana R, Peiris S, Gerlach R, Allen DE.
Modelling and Forecasting Stock Price Movements with Serially Dependent Determinants. *Risks*. 2018; 6(2):52.
https://doi.org/10.3390/risks6020052

**Chicago/Turabian Style**

Yatigammana, Rasika, Shelton Peiris, Richard Gerlach, and David Edmund Allen.
2018. "Modelling and Forecasting Stock Price Movements with Serially Dependent Determinants" *Risks* 6, no. 2: 52.
https://doi.org/10.3390/risks6020052