# Modeling Autoregressive Processes with Moving-Quantiles-Implied Nonlinearity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model with MQs and Its Functional-Coefficient Form

## 3. Some Characterizations of the MQ Process

#### 3.1. Stationarity and Existence of Moments

**Proposition 1.**

**Remark 2.**

**Proposition 2.**

#### 3.2. Estimation

**Proposition 3.**

#### 3.3. Test for the MQ Terms

**Theorem 1.**

**Remark 3.**

#### 3.4. Finite Sample Properties of the Parameter Estimator

**Figure 1.**The bias and the mean squared error of the OLS estimator as in Equation (5) of parameters of the autoregressive moving sample quantiles (AR-MQ) model with the first-order linear autoregressive and the moving maximum or moving median term from the indicated window sizes. Colors identify different parameters under investigation: red, autoregressive parameter in a pure autoregression (AR in AR); blue, MQ term in a pure MQ model (MQ in MQ); black, autoregressive parameter in the AR-MQ model (AR in AR-MQ); green, MQ term in the AR-MQ model (MQ in AR-MQ). The solid and dashed lines (both blue and green) are used to identify the moving maximum and moving median, respectively.

#### 3.5. Finite Sample Properties of the MQ Tests

**Figure 2.**Distribution of the empirical sizes of the tests under ${H}_{0}:\theta =0$. The red line corresponds to the unconstrained estimation of Equation (1) (as in Theorem 1), whereas the blue line corresponds to the estimation with the constrained linear autoregressive part (as in Remark 3).

**Figure 3.**Distribution of the empirical sizes of the test (as in Theorem 1) under ${H}_{1}:\theta \ne 0$. Each of the four figures represents the results with different sample sizes, where the five lines correspond to the presence of different quartiles in the data generating process (DGP) (minimum, first quartile, median, third quartile and maximum).

**Figure 4.**Distribution of empirical sizes under ${H}_{1}:\theta \ne 0$ for the test with the constrained linear autoregressive parameters (as in Remark 3). Each of the four figures represents the results with different sample sizes, where the five lines correspond to the presence of different quartiles in the DGP (minimum, first quartile, median, third quartile and maximum).

#### 3.6. Simulation Evidence on the Power against Other Non-Linearities

**L**contains various short and long memory linear stationary processes. Block

**N**contains several non-linear processes taken from 4 [42,43]. We augment this set of models with an additional one in the spirit of [44], where the unconditional moment is driven by a bounded cyclical deterministic function. Many of the considered models are well known to be capable of generating pseudo long memory features, such as slowly decaying sample autocorrelation function (see, e.g., [43]).

**L**. It reveals that the empirical sizes track very closely the nominal ones, especially at the significance levels that are usually used in practice. Similar results (unreported) hold for the RESET tests, apart from the AR(20) case. This happens because in the testing, we fixed $k=12$, and the RESET test, being a general specification test, rejects the hypothesis of zero conditional expectation of errors in the misspecified model.

Block | Code | Type of Model | DGP | Power Observed in Simulations: | ||
---|---|---|---|---|---|---|

MQ Test | RESET Tests (Any) | MQ More Powerful? | ||||

Linear | L1 | AR(1) | ${x}_{t}=0.8{x}_{t-1}+{\epsilon}_{t}.$ | – | – | – |

L2 | AR(2) | ${x}_{t}=0.5{x}_{t-1}+0.3{x}_{t-2}+{\epsilon}_{t}.$ | – | – | – | |

L3 | ARMA (1,1) | ${x}_{t}=0.8{x}_{t-1}+{\epsilon}_{t}-0.5{\epsilon}_{t-1}.$ | – | – | – | |

L4 | AR(12) | ${x}_{t}={\sum}_{i=1}^{12}{\varphi}_{i}{x}_{t-i}+{\epsilon}_{t},\phantom{\rule{4pt}{0ex}}{\varphi}_{i}=0.8/12,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,12.$ | – | – | – | |

L5 | AR(20) | ${x}_{t}={\sum}_{i=1}^{20}{\varphi}_{i}{x}_{t-i}+{\epsilon}_{t},\phantom{\rule{4pt}{0ex}}{\varphi}_{i}=0.8/20,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,20.$ | – | + | – | |

L6 | FARIMA (0,d,0) | ${\Delta}^{d}{x}_{t}={\epsilon}_{t},\phantom{\rule{4pt}{0ex}}d=0.2.$ | – | – | – | |

L7 | FARIMA(0,d,0) | ${\Delta}^{d}{x}_{t}={\epsilon}_{t},\phantom{\rule{4pt}{0ex}}d=0.4.$ | – | – | – | |

L8 | FARIMA(1,d,1) | ${\Delta}^{d}{x}_{t}=0.8{\Delta}^{d}{x}_{t-1}+{\epsilon}_{t}-0.5{\epsilon}_{t-1},\phantom{\rule{4pt}{0ex}}d=0.2.$ | – | – | – | |

L9 | FARIMA(1,d,1) | ${\Delta}^{d}{x}_{t}=0.8{\Delta}^{d}{x}_{t-1}+{\epsilon}_{t}-0.5{\epsilon}_{t-1},\phantom{\rule{4pt}{0ex}}d=0.4.$ | – | – | – | |

Non-linear | N1 | SETAR | ${x}_{t}=0.8{x}_{t-1}({1\phantom{\rule{-2.9pt}{0ex}}\mathrm{I}}_{{x}_{t-1}<0}-{1\phantom{\rule{-2.9pt}{0ex}}\mathrm{I}}_{{x}_{t-1}>0})+{\epsilon}_{t}.$ | + | + | – |

N2 | ESTAR | ${x}_{t}=0.8{x}_{t-1}\left(1-1.5(1-{e}^{-{x}_{t-1}^{2}})\right)+{\epsilon}_{t}.$ | + | + | – | |

N3 | LSTAR | ${x}_{t}=0.8{x}_{t-1}\left(1-1.5/(1+{e}^{-{x}_{t-1}})\right)+{\epsilon}_{t}.$ | + | + | – | |

N4 | BL | ${x}_{t}=0.8{x}_{t-1}(1+0.375{\epsilon}_{t-1})+{\epsilon}_{t}.$ | + | + | – | |

N5 | MQ(q${}_{0.5}$) | ${x}_{t}=0.3{x}_{t-1}+0.5{q}_{0.5}\left(12\right)+{\epsilon}_{t}.$ | + | – | + | |

N6 | MQ(q${}_{0.75}$) | ${x}_{t}=0.3{x}_{t-1}+0.5{q}_{0.75}\left(12\right)+{\epsilon}_{t}.$ | + | – | + | |

N7 | MQ(q${}_{1}$) | ${x}_{t}=0.3{x}_{t-1}+0.5{q}_{1}\left(12\right)+{\epsilon}_{t}.$ | + | – | + | |

N8 | MS -GARCH | ${x}_{t}=log\left({r}_{t}^{2}\right),\phantom{\rule{4pt}{0ex}}{r}_{t}=\sqrt{{h}_{t}}{\epsilon}_{t},\phantom{\rule{4pt}{0ex}}{h}_{t}=1+3{s}_{t}+0.4{r}_{t-1}^{2}+0.3{h}_{t-1}.$ | + | + | + | |

N9 | MS-AR | ${x}_{t}=0.8{x}_{t-1}(2{S}_{t}-1)+{\epsilon}_{t}.$ | – | + | – | |

N10 | MS-mean | ${x}_{t}=2{s}_{t}-1+{\epsilon}_{t}.$ | – | + | – | |

N11 | RLS -NS | ${x}_{t}={\mu}_{t}+\sqrt{5}{\epsilon}_{t},\phantom{\rule{4pt}{0ex}}{\mu}_{t}={\mu}_{t-1}+{j}_{t}{\eta}_{t},\phantom{\rule{4pt}{0ex}}{j}_{t}\sim i.i.d.B(1,0.00001)$ | – | + | – | |

N12 | RLS-S | ${x}_{t}={\mu}_{t}+{\epsilon}_{t},\phantom{\rule{4pt}{0ex}}{\mu}_{t}=(1-{j}_{t}){\mu}_{t-1}+{j}_{t}{\eta}_{t},\phantom{\rule{4pt}{0ex}}{j}_{t}\sim i.i.d.B(1,0.003)$ | – | + | – | |

N13 | Trend-power | ${x}_{t}={f}_{t}+{\epsilon}_{t},\phantom{\rule{4pt}{0ex}}{f}_{t}=3{t}^{-0.1}$ | – | – | – | |

N14 | Trend-cycles | ${x}_{t}={c}_{t}+{\epsilon}_{t},\phantom{\rule{4pt}{0ex}}{c}_{t}=0.3sin(3\pi t/n)+sin(6\pi t/n),\phantom{\rule{4pt}{0ex}}t\in \{1,2,\dots ,n\},n=2,000.$ | – | – | – |

**Figure 5.**Distribution of empirical sizes of the MQ test for the data generated from the stationary short and long memory linear processes (DGPs from the

**L**block in Table 1).

#### 3.7. Some Empirical Features of Realizations of the AR-MQ Process

**Figure 6.**The realizations of the AR-MQ model (with the moving median from the indicated window sizes k), their sample autocorrelation functions (ACFs) (

**middle**) and the ACFs of the linear autoregressive (AR) processes obtained by restricting the parameter of the MQ part to zero in the AR-MQ model (

**right**).

## 4. Empirical Application

#### 4.1. Significance of the MQ Terms

**Table 2.**p-values of the test for the absence of moving quantile (MQ) effects (estimation sample: 1–2,000). HAR, heterogeneous autoregression model.

Models | MQ Window | Quantile Structures: | |||
---|---|---|---|---|---|

Min-Med-Max | Quartiles | Quintiles | Deciles | ||

AR(12) | 12 | $0.005\phantom{\rule{4pt}{0ex}}\left(0.006\right)$ | $0.021\phantom{\rule{4pt}{0ex}}\left(0.027\right)$ | $0.022\phantom{\rule{4pt}{0ex}}\left(0.027\right)$ | $0.018\phantom{\rule{4pt}{0ex}}\left(0.030\right)$ |

20 | $0.052\phantom{\rule{4pt}{0ex}}\left(0.044\right)$ | $0.083\phantom{\rule{4pt}{0ex}}\left(0.082\right)$ | $0.109\phantom{\rule{4pt}{0ex}}\left(0.102\right)$ | $0.226\phantom{\rule{4pt}{0ex}}\left(0.271\right)$ | |

AR(20) | 12 | $0.006\phantom{\rule{4pt}{0ex}}\left(0.008\right)$ | $0.024\phantom{\rule{4pt}{0ex}}\left(0.033\right)$ | $0.026\phantom{\rule{4pt}{0ex}}\left(0.033\right)$ | $0.016\phantom{\rule{4pt}{0ex}}\left(0.030\right)$ |

20 | $0.022\phantom{\rule{4pt}{0ex}}\left(0.023\right)$ | $0.081\phantom{\rule{4pt}{0ex}}\left(0.086\right)$ | $0.085\phantom{\rule{4pt}{0ex}}\left(0.094\right)$ | $0.229\phantom{\rule{4pt}{0ex}}\left(0.292\right)$ | |

HAR | 12 | $0.004\phantom{\rule{4pt}{0ex}}\left(0.007\right)$ | $0.015\phantom{\rule{4pt}{0ex}}\left(0.030\right)$ | $0.020\phantom{\rule{4pt}{0ex}}\left(0.040\right)$ | $0.010\phantom{\rule{4pt}{0ex}}\left(0.020\right)$ |

20 | $0.010\phantom{\rule{4pt}{0ex}}\left(0.011\right)$ | $0.040\phantom{\rule{4pt}{0ex}}\left(0.047\right)$ | $0.044\phantom{\rule{4pt}{0ex}}\left(0.055\right)$ | $0.137\phantom{\rule{4pt}{0ex}}\left(0.202\right)$ | |

ALMON(12) | 12 | $0.001\phantom{\rule{4pt}{0ex}}\left(0.021\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.085\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.098\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.071\right)$ |

20 | $0.001\phantom{\rule{4pt}{0ex}}\left(0.020\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.072\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.083\right)$ | $0.002\phantom{\rule{4pt}{0ex}}\left(0.268\right)$ | |

ALMON(20) | 12 | $0.001\phantom{\rule{4pt}{0ex}}\left(0.021\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.085\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.097\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.071\right)$ |

20 | $0.001\phantom{\rule{4pt}{0ex}}\left(0.020\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.072\right)$ | $0.001\phantom{\rule{4pt}{0ex}}\left(0.083\right)$ | $0.003\phantom{\rule{4pt}{0ex}}\left(0.267\right)$ |

#### 4.2. In-Sample Performance and Forecasting Precision

**Figure 8.**Effect of the moving window size, which is used to calculate the MQs, on the values of the information criteria for the ALMON(12) model with all MQs (red line) and only including the moving median (black line).

Sample: | 1–1,000 | 1–2,000 | ||
---|---|---|---|---|

Criterion: | AIC | BIC | AIC | BIC |

HAR | 161.1 | 185.5 | 422.7 | 450.6 |

AR(12) | 167.2 | 235.8 | 423.2 | 501.5 |

AR(20) | 175.7 | 283.2 | 432.0 | 555.0 |

ALMON(12) | 167.2 | 186.8 | 435.8 | 458.2 |

ALMON(20) | 162.8 | 182.4 | 429.9 | 452.3 |

HAR-MQ(12) | 158.8 | 188.2 | 414.6 | 448.2 |

HAR-MQ(20) | 162.3 | 191.7 | 418.7 | 452.3 |

AR(12)-MQ(12) | 167.5 | 240.9 | 416.2 | 500.2 |

AR(12)-MQ(20) | 164.2 | 237.6 | 417.8 | 501.7 |

AR(20)-MQ(12) | 176.2 | 288.6 | 425.5 | 554.1 |

AR(20)-MQ(20) | 177.1 | 289.5 | 429.2 | 557.8 |

ALMON(12)-MQ(12) | 183.2 | 207.7 | 466.1 | 494.1 |

ALMON(20)-MQ(12) | 179.3 | 203.7 | 461.9 | 489.8 |

ALMON(12)-MQ(20) | 158.1 | 182.5 | 411.3 | 439.2 |

ALMON(20)-MQ(20) | 212.0 | 236.4 | 522.3 | 550.3 |

**Table 4.**Relative out-of-sample forecasting precision (the benchmark is the mean squared forecasting error of the HAR model in each case). Index: S&P 500 (live). Initial series: realized variance (RV). Transformation: $log\left(100\sqrt{252R{V}_{t}}\right)$. Forecasting horizon: one day.

Initial Estimation Sample: | 1–1,000 | 1–1,000 | 1–2,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 1,001–2,000 | 1,001–3,000 | 2,001–3,000 | ||||||

Type of Forecast: | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 1.002 | 1.001 | 1.001 | 0.999 | 0.995 | 0.996 | 0.993 | 0.990 * | 0.989 * |

AR(20) | 1.008 | 1.007 | 1.012 | 1.002 | 0.998 | 1.003 | 0.990 * | 0.990 | 0.993 |

ALMON(12) | 1.018 | 1.006 | 0.999 | 1.001 | 0.995 | 0.993 | 0.987 * | 0.986 * | 0.986 * |

ALMON(20) | 1.014 | 1.005 | 0.998 | 0.999 | 0.994 | 0.992 * | 0.987 * | 0.986 * | 0.986 * |

HAR-MQ(12) | 0.996 | 0.995 | 0.996 | 1.002 | 1.001 | 1.001 | 1.009 | 1.005 | 1.005 |

HAR-MQ(20) | 0.996 ** | 0.996 * | 0.995 | 0.997 ** | 0.999 | 0.999 | 1.001 | 1.001 | 1.002 |

AR(12)-MQ(12) | 0.995 | 0.995 | 0.995 | 0.999 | 0.995 | 0.998 | 1.003 | 0.995 | 0.995 |

AR(12)-MQ(20) | 0.998 | 0.999 | 0.999 | 0.997 | 0.993 * | 0.995 | 0.992 | 0.989 * | 0.988 * |

AR(20)-MQ(12) | 1.003 | 1.001 | 1.007 | 1.002 | 0.998 | 1.005 | 1.000 | 0.996 | 0.999 |

AR(20)-MQ(20) | 1.004 | 1.004 | 1.009 | 1.000 | 0.997 | 1.003 | 0.991 | 0.991 | 0.995 |

ALMON(12)-MQ(12) | 0.994 | 0.990 | 0.987 ** | 0.994 | 0.991 * | 0.988 ** | 0.996 | 0.991 | 0.989 |

ALMON(20)-MQ(12) | 0.993 | 0.990 | 0.987 ** | 0.993 | 0.990 * | 0.987 ** | 0.996 | 0.991 | 0.990 |

ALMON(12)-MQ(20) | 0.992 | 0.992 | 0.988 ** | 0.987 *** | 0.987 *** | 0.985 *** | 0.985 ** | 0.983 ** | 0.983 ** |

ALMON(20)-MQ(20) | 0.992 | 0.992 | 0.988 ** | 0.987 *** | 0.987 *** | 0.984 *** | 0.985 ** | 0.983 ** | 0.983 ** |

## 5. Robustness and Extensions

#### 5.1. Sensitivity of S&P 500 Analysis

#### 5.2. Relevance of Larger MQ Windows for S&P 500

Coefficients | Extensions | |||
---|---|---|---|---|

none | q${}_{0.75}$(60) | q${}_{1}$(120) | ||

Intercept | $0.1239$ *** | $0.0812$ * | $0.0664$ | |

$\left(0.0405\right)$ | $\left(0.0459\right)$ | $\left(0.0451\right)$ | ||

of ALMON restriction: | ${\psi}_{0}$ | $0.7205$ *** | $0.7317$ *** | $0.7222$ *** |

$\left(0.0469\right)$ | $\left(0.0484\right)$ | $\left(0.0487\right)$ | ||

${\psi}_{1}$ | –$0.5960$ *** | –$0.5814$ *** | –$0.5813$ *** | |

$\left(0.0701\right)$ | $\left(0.0605\right)$ | $\left(0.0710\right)$ | ||

of MQ terms: | ||||

${q}_{0.5}\left(20\right)$ (moving median, window=20) | ${\theta}_{1}$ | $0.2310$ *** | $0.1478$ ** | $0.1711$ *** |

$\left(0.0476\right)$ | $\left(0.0621\right)$ | $\left(0.0532\right)$ | ||

${q}_{0.75}\left(60\right)$ (moving 3rd quartile, window=60) | ${\theta}_{2}$ | $0.0824$ ** | ||

$\left(0.0381\right)$ | ||||

${q}_{1}\left(120\right)$ (moving maximum, window=120) | ${\theta}_{3}$ | $0.0592$ *** | ||

$\left(0.0213\right)$ | ||||

Standard error of residuals | 0.2680 | 0.2685 | 0.2668 | |

Degrees of freedom | 1,977 | 1,936 | 1,876 | |

AIC | 411.279 | 411.289 | 375.117 | |

BIC | 439.236 | 444.715 | 408.354 | |

${\sum}_{i=1}^{3}|{\theta}_{i}|+{\sum}_{j=1}^{12}\left|{\beta}_{j}\right|$ | 0.9514 | 0.9619 | 0.9524 |

#### 5.3. Out-of-Sample Forecasting Performance (for More Indices)

**Table 6.**Relative out-of-sample forecasting precision (a benchmark is the mean squared forecast error of the HAR model in each case). Indices: seven different. Initial series: realized variance (RV). Transformation: $log\left(100\sqrt{252R{V}_{t}}\right)$. Forecasting horizon: one day. Forecasting type: rolling. Initial estimation sample: 1–2,000.

Models | Indices: | ||||||
---|---|---|---|---|---|---|---|

S&P 500 | FTSE 100 | Nikkei 225 | DAX | Russell 2000 | AORD | DJIA | |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 0.989 * | 0.994 | 0.993 | 0.995 | 0.984 * | 1.002 | 0.991 |

AR(20) | 0.993 | 0.997 | 0.996 | 0.994 | 0.985 | 1.002 | 0.998 |

ALMON(12) | 0.986 * | 1.018 | 0.991 | 1.003 | 0.997 | 0.99 * | 0.985 ** |

ALMON(20) | 0.986 * | 1.019 | 0.991 | 1.003 | 0.996 | 0.988 ** | 0.985 ** |

HAR-MQ(12) | 1.005 | 1.000 | 1.000 | 1.002 | 1.006 | 1.001 | 1.003 |

HAR-MQ(20) | 1.002 | 1.001 | 0.996 | 0.998 | 1.000 | 1.001 | 0.999 |

AR(12)-MQ(12) | 0.995 | 0.995 | 0.994 | 0.995 | 0.990 | 1.003 | 0.995 |

AR(12)-MQ(20) | 0.988 * | 0.993 | 0.989 ** | 0.992 | 0.979 ** | 0.997 | 0.99 * |

AR(20)-MQ(12) | 0.999 | 0.998 | 0.997 | 0.995 | 0.993 | 1.002 | 1.002 |

AR(20)-MQ(20) | 0.995 | 0.997 | 0.992 | 0.992 | 0.985 | 1.003 | 0.997 |

ALMON(12)-MQ(12) | 0.989 | 1.004 | 0.982 ** | 0.991 | 0.986 * | 0.991 | 0.990 |

ALMON(20)-MQ(12) | 0.990 | 1.004 | 0.982 ** | 0.990 | 0.986 * | 0.988 ** | 0.990 |

ALMON(12)-MQ(20) | 0.983 ** | 1.006 | 0.975 *** | 0.987 * | 0.979 *** | 0.984 *** | 0.983 *** |

ALMON(20)-MQ(20) | 0.983 ** | 1.006 | 0.975 *** | 0.987 * | 0.979 *** | 0.987 ** | 0.983 *** |

ALMON(12)-MQ(20:0.5; 60:0.75) | 0.982 ** | 1.006 | 0.978 *** | 0.987 * | 0.976 *** | 0.983 ** | 0.984 ** |

ALMON(20)-MQ(20:0.5; 60:0.75) | 0.982 ** | 1.006 | 0.978 *** | 0.987 * | 0.975 *** | 0.985 ** | 0.984 ** |

ALMON(12)-MQ(20:0.5; 120:1) | 0.980 ** | 1.006 | 0.978 *** | 0.985 ** | 0.975 *** | 0.985 ** | 0.980 *** |

ALMON(20)-MQ(20:0.5; 120:1) | 0.981 ** | 1.006 | 0.978 *** | 0.986 ** | 0.975 *** | 0.986 ** | 0.981 *** |

#### 5.4. MQ and RESET Testing Results Using the Latest Available Dataset (for October 8, 2014)

**Table 7.**Heteroscedasticity robust p-values of the MQ and RESET testing (for neglected non-linearity).

AR Order Under H0 | Test Specification | Indices: | ||||||
---|---|---|---|---|---|---|---|---|

S&P 500 | FTSE 100 | Nikkei 225 | DAX | Russell 2000 | AORD | DJIA | ||

$k=12$ | MQ(20: 0,0.5,1) | 0.002 | 0.002 | 0.001 | 0.000 | 0.002 | 0.003 | 0.001 |

MQ(60: 0,0.5,1) | 0.001 | 0.000 | 0.010 | 0.001 | 0.000 | 0.000 | 0.004 | |

MQ(120: 0,0.5,1) | 0.000 | 0.000 | 0.001 | 0.000 | 0.001 | 0.000 | 0.000 | |

MQ(20: 0,0.25,0.5,0.75,1) | 0.005 | 0.000 | 0.006 | 0.000 | 0.002 | 0.010 | 0.001 | |

MQ(60: 0,0.25,0.5,0.75,1) | 0.005 | 0.000 | 0.008 | 0.002 | 0.002 | 0.000 | 0.013 | |

MQ(120: 0,0.25,0.5,0.75,1) | 0.000 | 0.000 | 0.003 | 0.003 | 0.006 | 0.000 | 0.001 | |

RESET(2) | 0.048 | 0.863 | 0.348 | 0.068 | 0.000 | 0.248 | 0.018 | |

RESET(2:3) | 0.040 | 0.045 | 0.524 | 0.014 | 0.001 | 0.311 | 0.005 | |

RESET(2:4) | 0.080 | 0.090 | 0.691 | 0.031 | 0.002 | 0.466 | 0.011 | |

$k=20$ | MQ(20: 0,0.5,1) | 0.012 | 0.009 | 0.298 | 0.001 | 0.022 | 0.535 | 0.007 |

MQ(60: 0,0.5,1) | 0.035 | 0.001 | 0.121 | 0.132 | 0.024 | 0.000 | 0.071 | |

MQ(120: 0,0.5,1) | 0.001 | 0.003 | 0.012 | 0.026 | 0.037 | 0.004 | 0.002 | |

MQ(20: 0,0.25,0.5,0.75,1) | 0.032 | 0.031 | 0.548 | 0.007 | 0.042 | 0.667 | 0.008 | |

MQ(60: 0,0.25,0.5,0.75,1) | 0.076 | 0.005 | 0.085 | 0.167 | 0.087 | 0.002 | 0.133 | |

MQ(120: 0,0.25,0.5,0.75,1) | 0.003 | 0.012 | 0.031 | 0.084 | 0.103 | 0.003 | 0.008 | |

RESET(2) | 0.055 | 0.707 | 0.304 | 0.063 | 0.001 | 0.249 | 0.025 | |

RESET(2:3) | 0.058 | 0.106 | 0.491 | 0.029 | 0.002 | 0.353 | 0.010 | |

RESET(2:4) | 0.112 | 0.191 | 0.660 | 0.061 | 0.005 | 0.515 | 0.024 |

## 6. Final Remarks

## Acknowledgments

## Author Contributions

## Appendix

## A. Proofs

**Proof of Proposition 1.**

**Proof of Proposition 2.**

**Proof of Proposition 3.**

**Proof of Theorem 1.**

**Proof of Remark 3.**

## B. Additional Results from the Simulations

**N**block in Table 1).

**N**block in Table 1).

**Figure B1.**The Q–Qplots (sample quantiles versus the theoretical ones) of 2,000 realizations of the AR-MQ model (with the moving maximum from the indicated window sizes k) and the (non-parametrically estimated) density of the standardized realization of the case with $k=60$ (

**bottom right**).

**Figure B2.**Distribution of empirical sizes of the MQ test for the data generated by the non-linear processes N1–N8 (DGPs from the

**N**block in Table 1).

**Figure B3.**Distribution of empirical sizes of the MQ test for the data generated by the non-linear processes N9–N14 (DGPs from the

**N**block in Table 1).

## C. Examples of Estimated Models for the S&P 500 Index Returns (Estimation Sample: 1–2,000)

Coefficients | Models | ||||
---|---|---|---|---|---|

HAR | HAR-MQ(20) | ALMON(12) | ALMON(12)-MQ(20) | ||

Intercept | $0.1399$ *** | $0.1249$ *** | $0.1788$ *** | $0.1239$ *** | |

$\left(0.0402\right)$ | $\left(0.0396\right)$ | $\left(0.0422\right)$ | $\left(0.0405\right)$ | ||

of HAR restriction: | ${\psi}_{1}$ | $0.2541$ *** | $0.2535$ *** | ||

$\left(0.0308\right)$ | $\left(0.0309\right)$ | ||||

${\psi}_{2}$ | $0.4712$ *** | $0.4755$ *** | |||

$\left(0.0508\right)$ | $\left(0.0503\right)$ | ||||

${\psi}_{3}$ | $0.2196$ *** | $-0.1231$ | |||

$\left(0.0423\right)$ | $\left(0.1555\right)$ | ||||

of ALMON restriction: | ${\psi}_{1}$ | $0.9296$ *** | $0.7205$ *** | ||

$\left(0.0167\right)$ | $\left(0.0469\right)$ | ||||

${\psi}_{2}$ | –$0.3965$ *** | –$0.5960$ *** | |||

$\left(0.0321\right)$ | $\left(0.0701\right)$ | ||||

of moving median | θ | $0.3453$ ** | $0.2310$ *** | ||

$\left(0.1501\right)$ | $\left(0.0476\right)$ | ||||

Standard error of residuals | 0.2688 | 0.2685 | 0.2697 | 0.2680 | |

Degrees of freedom | 1976 | 1975 | 1985 | 1977 | |

AIC | 422.655 | 418.750 | 435.775 | 411.279 | |

BIC | 450.609 | 452.295 | 458.155 | 439.236 | |

$\left|\theta \right|+{\sum}_{j=1}^{k}\left|{\varphi}_{j}\right|$ | 0.9448 | 1.1358 | 0.9296 | 0.9514 |

**Table C2.**Prevalence of the sufficient stability condition defined in Proposition 1 for the estimated models with MQ terms.

Estimation Sample | |||
---|---|---|---|

1–1,000 | 1–2,000 | 1–3,000 | |

HAR-MQ(12) | + | + | + |

HAR-MQ(20) | + | – | – |

AR(12)-MQ(12) | – | – | – |

AR(12)-MQ(20) | + | + | + |

AR(20)-MQ(12) | – | – | – |

AR(20)-MQ(20) | – | – | – |

ALMON(12)-MQ(12) | + | + | + |

ALMON(20)-MQ(12) | + | + | + |

ALMON(12)-MQ(20) | + | + | + |

ALMON(20)-MQ(20) | + | + | + |

**Figure C1.**The empirical autocorrelation functions (ACF) of the residuals of the estimated models: HAR (

**left-top**), HAR-MQ(20) (

**right-top**), ALMON(12) (

**left-bottom**) and ALMON(12)-MQ(20) (

**right-bottom**).

## D. Out-of-Sample Forecasting Precision: MAPE (Table D1) and MASE (Table D2)

**Table D1.**Relative out-of-sample forecasting precision (the benchmark is the mean absolute percentage error (

**MAPE**) of the HAR model in each case). Index: S&P 500 (live). Initial series: realized variance (RV). Transformation: $log\left(100\sqrt{252R{V}_{t}}\right)$. Forecasting horizon: one day.

Initial Estimation Sample: | 1–1,000 | 1–1,000 | 1–2,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 1,001–2,000 | 1,001–3,000 | 2,001–3,000 | ||||||

Type of Forecast: | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 1.000 | 1.000 | 1.001 | 1.001 | 0.997 | 0.997 | 0.997 | 0.993 * | 0.992 |

AR(20) | 1.001 | 1.001 | 1.001 | 1.001 | 0.997 | 1.000 | 0.995 | 0.993 ** | 0.995 |

ALMON(12) | 1.012 | 1.003 | 0.999 | 1.003 | 0.997 | 0.995 * | 0.990 | 0.991 * | 0.990 * |

ALMON(20) | 1.009 | 1.001 | 0.998 | 1.001 | 0.996 | 0.994 * | 0.991 | 0.991 * | 0.991 * |

HAR-MQ(12) | 0.997 * | 0.997 * | 0.998 | 1.000 | 0.999 | 0.999 | 1.004 | 1.002 | 1.001 |

HAR-MQ(20) | 0.998 | 0.999 | 0.999 | 0.998 * | 0.999 | 0.999 | 0.999 | 0.999 | 0.998 |

AR(12)-MQ(12) | 0.996 | 0.996 | 0.998 | 0.998 | 0.995 * | 0.996 | 1.000 | 0.994 | 0.992 |

AR(12)-MQ(20) | 0.997 | 0.999 | 1.000 | 0.999 | 0.996 * | 0.996 | 0.995 | 0.992 ** | 0.992 ** |

AR(20)-MQ(12) | 0.998 | 0.998 | 0.998 | 1.000 | 0.996 | 0.999 | 0.998 | 0.993 | 0.996 |

AR(20)-MQ(20) | 0.999 | 1.001 | 1.000 | 0.999 | 0.995 | 0.999 | 0.992 | 0.989 ** | 0.991 * |

ALMON(12)-MQ(12) | 0.996 | 0.994 * | 0.994 * | 0.995 * | 0.993 ** | 0.992 ** | 0.994 | 0.991 * | 0.990 ** |

ALMON(20)-MQ(12) | 0.995 * | 0.994 * | 0.994 * | 0.994 ** | 0.992 ** | 0.992 ** | 0.994 | 0.991 * | 0.990 ** |

ALMON(12)-MQ(20) | 0.993 * | 0.995 | 0.995 * | 0.990 *** | 0.992 *** | 0.992 *** | 0.987 *** | 0.988 *** | 0.988 *** |

ALMON(20)-MQ(20) | 0.993 * | 0.995 | 0.995 * | 0.990 *** | 0.992 *** | 0.992 *** | 0.989 *** | 0.988 *** | 0.988 *** |

**Table D2.**Relative out-of-sample forecasting precision (the benchmark is the mean absolute scaled error (

**MASE**) of the HAR model in each case). Index: S&P 500 (live). Initial series: realized variance (RV). Transformation: $log\left(100\sqrt{252R{V}_{t}}\right)$. Forecasting horizon: one day.

Initial Estimation Sample: | 1–1,000 | 1–1,000 | 1–2,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 1,001–2,000 | 1,001–3,000 | 2,001–3,000 | ||||||

Type of Forecast: | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 0.998 | 1.000 | 1.001 | 1.000 | 0.996 | 0.996 | 0.997 | 0.993 | 0.991 |

AR(20) | 0.999 | 1.002 | 1.002 | 1.000 | 0.997 | 0.999 | 0.995 | 0.992 | 0.994 |

ALMON(12) | 1.006 | 1.000 | 0.997 | 0.999 | 0.996 | 0.995 | 0.993 | 0.992 | 0.992 |

ALMON(20) | 1.004 | 0.999 | 0.996 | 0.998 | 0.995 | 0.994 | 0.993 | 0.992 | 0.992 |

HAR-MQ(12) | 0.996 | 0.996 | 0.998 | 1.000 | 1.000 | 1.000 | 1.005 | 1.003 | 1.002 |

HAR-MQ(20) | 0.999 | 0.999 | 0.999 | 0.999 | 1.000 | 1.000 | 1.001 | 1.000 | 1.000 |

AR(12)-MQ(12) | 0.994 | 0.995 | 0.997 | 0.999 | 0.995 | 0.996 | 1.002 | 0.995 | 0.993 |

AR(12)-MQ(20) | 0.997 | 1.000 | 1.001 | 0.999 | 0.996 | 0.996 | 0.996 | 0.992 | 0.991 |

AR(20)-MQ(12) | 0.996 | 0.998 | 0.998 | 1.000 | 0.996 | 1.000 | 1.000 | 0.994 | 0.996 |

AR(20)-MQ(20) | 0.998 | 1.001 | 1.001 | 0.999 | 0.995 | 0.999 | 0.994 | 0.990 | 0.991 |

ALMON(12)-MQ(12) | 0.994 | 0.993 | 0.993 | 0.994 | 0.993 | 0.992 | 0.996 | 0.993 | 0.991 |

ALMON(20)-MQ(12) | 0.993 | 0.992 | 0.992 | 0.993 | 0.992 | 0.992 | 0.996 | 0.993 | 0.992 |

ALMON(12)-MQ(20) | 0.993 | 0.995 | 0.995 | 0.990 | 0.992 | 0.991 | 0.989 | 0.988 | 0.988 |

ALMON(20)-MQ(20) | 0.993 | 0.995 | 0.995 | 0.990 | 0.992 | 0.991 | 0.989 | 0.988 | 0.988 |

## E. Robustness Analysis

**Table E1.**Relative out-of-sample forecasting precision (the benchmark is the mean squared forecast error of the HAR model in each case). Index: S&P 500 (live).

**Initial series: realized kernel (RK)**. Transformation: $log\left(100\sqrt{252R{K}_{t}}\right)$. Forecasting horizon: one day.

Initial Estimation Sample: | 1–1,000 | 1–1,000 | 1–2,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 1,001–2,000 | 1,001–3,000 | 2,001–3,000 | ||||||

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 0.999 | 0.999 | 1.000 | 0.998 | 0.995 | 0.997 | 0.992 | 0.991 | 0.990 |

AR(20) | 1.006 | 1.005 | 1.009 | 1.000 | 0.998 | 1.003 | 0.989 * | 0.991 | 0.993 |

ALMON(12) | 1.020 | 1.008 | 1.002 | 1.004 | 0.998 | 0.998 | 0.988 | 0.988 | 0.988 |

ALMON(20) | 1.017 | 1.007 | 1.001 | 1.003 | 0.997 | 0.997 | 0.989 | 0.989 | 0.988 |

HAR-MQ(12) | 0.998 | 0.998 | 0.998 | 1.003 | 1.001 | 1.001 | 1.007 | 1.004 | 1.004 |

HAR-MQ(20) | 1.000 | 0.999 | 0.998 | 1.000 | 0.999 | 1.000 | 0.999 | 1.000 | 1.000 |

AR(12)-MQ(12) | 0.992 | 0.995 | 0.997 | 0.997 | 0.995 | 0.998 | 1.001 | 0.996 | 0.996 |

AR(12)-MQ(20) | 1.001 | 0.999 | 1.001 | 0.999 | 0.994 | 0.997 | 0.991 * | 0.990 * | 0.989 * |

AR(20)-MQ(12) | 1.000 | 1.002 | 1.006 | 0.999 | 0.998 | 1.004 | 0.997 | 0.995 | 0.998 |

AR(20)-MQ(20) | 1.007 | 1.005 | 1.008 | 1.000 | 0.997 | 1.003 | 0.988 | 0.990 | 0.993 |

ALMON(12)-MQ(12) | 0.993 | 0.992 | 0.990 * | 0.993 | 0.991 * | 0.989 ** | 0.993 | 0.989 | 0.988 * |

ALMON(20)-MQ(12) | 0.992 | 0.992 | 0.989 * | 0.992 * | 0.991 * | 0.988 ** | 0.993 | 0.989 | 0.988 * |

ALMON(12)-MQ(20) | 0.993 | 0.993 | 0.991 * | 0.987 *** | 0.988 * | 0.987 *** | 0.983 ** | 0.983 ** | 0.982 ** |

ALMON(20)-MQ(20) | 0.993 | 0.993 | 0.991 * | 0.987 *** | 0.988 * | 0.986 *** | 0.983 ** | 0.983 ** | 0.982 ** |

**Table E2.**Relative out-of-sample forecasting precision (the benchmark is the mean squared forecast error of the HAR model in each case). Index: S&P 500 (live). Initial series: realized variance (RV). Transformation: $log\left(100\sqrt{252R{V}_{t}}\right)$.

**Forecasting horizon: one week**.

Initial Estimation Sample: | 1–1,000 | 1–1,000 | 1–2,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 1,001–2,000 | 1,001–3,000 | 2,001–3,000 | ||||||

Type of Forecast: | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 0.993 | 0.986 | 0.987 | 0.997 | 0.994 | 0.999 | 0.997 | 1.000 | 0.999 |

AR(20) | 0.995 | 0.988 | 0.998 | 0.998 | 0.993 | 1.005 | 0.992 | 0.997 | 0.998 |

ALMON(12) | 1.031 | 1.010 | 1.005 | 1.016 | 1.004 | 1.004 | 1.002 | 1.000 | 0.999 |

ALMON(20) | 1.015 | 1.003 | 0.998 | 1.008 | 1.001 | 0.999 | 1.002 | 1.000 | 0.999 |

HAR-MQ(12) | 0.992 | 0.984 * | 0.981 * | 1.001 | 0.998 | 0.997 | 1.013 | 1.008 | 1.008 |

HAR-MQ(20) | 0.995 ** | 0.991 ** | 0.983 ** | 0.995 *** | 0.993 | 0.992 | 0.996 | 0.995 | 0.998 |

AR(12)-MQ(12) | 0.986 | 0.977 | 0.978 | 0.993 | 0.992 | 0.998 | 1.002 | 1.002 | 1.004 |

AR(12)-MQ(20) | 0.988 | 0.979 * | 0.980 | 0.993 | 0.986 | 0.990 | 0.990 | 0.991 | 0.990 |

AR(20)-MQ(12) | 0.989 | 0.981 | 0.988 | 0.996 | 0.992 | 1.004 | 0.997 | 1.000 | 1.003 |

AR(20)-MQ(20) | 0.992 | 0.982 | 0.984 | 0.996 | 0.987 | 0.999 | 0.987 | 0.992 | 0.995 |

ALMON(12)-MQ(12) | 1.178 | 1.156 | 1.116 | 1.227 | 1.238 | 1.223 | 1.294 | 1.296 | 1.299 |

ALMON(20)-MQ(12) | 0.985 | 0.971 | 0.966 ** | 0.991 | 0.987 | 0.983 | 1.003 | 0.998 | 0.998 |

ALMON(12)-MQ(20) | 0.988 | 0.977 * | 0.973 ** | 0.987 | 0.980 * | 0.978 * | 0.982 | 0.982 | 0.981 |

ALMON(20)-MQ(20) | 0.988 | 0.977 * | 0.971 ** | 0.987 | 0.980 * | 0.977 ** | 0.982 | 0.982 | 0.982 |

**Table E3.**Relative out-of-sample forecasting precision (the benchmark is the mean squared forecast error of the HAR model in each case). Index: S&P 500 (live). Initial series: realized volatility (RV).

**Transformation:**$\mathbf{100}\sqrt{\mathbf{252}{\mathbf{RV}}_{\mathbf{t}}}$. Forecasting horizon: one day.

Initial Estimation Sample: | 1–1,000 | 1–1,000 | 1–2,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 1,001–2,000 | 1,001–3,000 | 2,001–3,000 | ||||||

Type of Forecast: | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 1.015 | 1.013 | 1.018 | 0.994 | 0.994 | 1.020 | 0.989 | 0.991 | 0.991 |

AR(20) | 1.026 | 1.021 | 1.037 | 1.003 | 1.010 | 1.067 | 0.991 | 1.008 | 1.015 |

ALMON(12) | 1.044 | 1.023 | 1.006 | 1.008 | 0.996 | 0.990 | 1.000 | 0.992 | 0.990 |

ALMON(20) | 1.037 | 1.020 | 1.004 | 1.006 | 0.996 | 0.990 | 1.000 | 0.992 | 0.991 |

HAR-MQ(12) | 0.996 | 0.995 | 0.997 | 1.029 | 1.009 | 1.007 | 1.030 | 1.011 | 1.010 |

HAR-MQ(20) | 0.988 *** | 0.993 * | 0.997 | 0.996 | 1.002 | 1.010 | 1.001 | 1.004 | 1.004 |

AR(12)-MQ(12) | 0.995 | 1.001 | 1.013 | 1.014 | 0.998 | 1.022 | 1.019 | 0.998 | 0.999 |

AR(12)-MQ(20) | 1.008 | 1.009 | 1.019 | 0.991 | 0.995 | 1.029 | 0.987 * | 0.992 | 0.993 |

AR(20)-MQ(12) | 1.012 | 1.011 | 1.032 | 1.027 | 1.014 | 1.066 | 1.023 | 1.015 | 1.022 |

AR(20)-MQ(20) | 1.014 | 1.014 | 1.035 | 0.998 | 1.013 | 1.079 | 0.991 | 1.012 | 1.020 |

ALMON(12)-MQ(12) | 1.009 | 1.016 | 1.020 | 1.056 | 1.045 | 1.038 | 1.067 | 1.050 | 1.049 |

ALMON(20)-MQ(12) | 0.989 | 1.019 | 1.025 | 1.010 | 0.996 | 0.999 | 1.066 | 0.992 | 0.990 |

ALMON(12)-MQ(20) | 0.996 | 1.002 | 1.046 | 0.982 ** | 0.987 | 0.985 | 0.981 ** | 0.984 | 0.981 * |

ALMON(20)-MQ(20) | 0.996 | 1.002 | 0.998 | 0.982 ** | 0.981 ** | 0.979 * | 0.981 ** | 0.977 ** | 0.976 ** |

**Table E4.**Relative out-of-sample forecasting precision (the benchmark is the mean squared forecasting error of the HAR model in each case). Index: S&P 500 (live). Initial series: realized variance (RV). Transformation: $log\left(100\sqrt{252R{V}_{t}}\right)$. Forecasting horizon: one day.

Initial Estimation Sample: | 1–125 | 1–250 | 1–500 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 126–3,000 | 251–3,000 | 501–3,000 | ||||||

Type of Forecast: | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 1.150 | 1.000 | 1.079 | 1.032 | 0.997 | 1.029 | 1.006 | 0.996 | 1.008 |

AR(20) | 1.121 | 1.008 | 1.172 | 1.062 | 1.003 | 1.070 | 1.059 | 1.003 | 1.031 |

ALMON(12) | 1.314 | 0.997 | 0.978 *** | 1.044 | 0.994 | 0.991 * | 1.019 | 0.993 | 0.992 * |

ALMON(20) | 1.259 | 0.996 | 0.982 *** | 1.045 | 0.994 | 0.990 ** | 1.016 | 0.993 | 0.991 ** |

HAR-MQ(12) | 0.982 *** | 1.000 | 1.006 | 1.002 | 1.001 | 1.004 | 1.012 | 1.002 | 1.004 |

HAR-MQ(20) | 1.076 | 1.000 | 1.015 | 1.010 | 0.999 | 1.003 | 0.998 | 0.999 | 0.999 |

AR(12)-MQ(12) | 1.124 | 1.001 | 1.081 | 1.029 | 0.997 | 1.034 | 1.005 | 0.996 | 1.009 |

AR(12)-MQ(20) | 1.396 | 1.001 | 1.093 | 1.057 | 0.997 | 1.034 | 1.032 | 0.996 | 1.010 |

AR(20)-MQ(12) | 1.132 | 1.008 | 1.179 | 1.059 | 1.004 | 1.076 | 1.057 | 1.004 | 1.031 |

AR(20)-MQ(20) | 1.401 | 1.009 | 1.200 | 1.094 | 1.004 | 1.075 | 1.062 | 1.003 | 1.030 |

ALMON(12)-MQ(12) | 1.065 | 0.991 * | 0.992 | 1.007 | 0.991* | 0.991** | 0.999 | 0.991 *** | 0.990 ** |

ALMON(20)-MQ(12) | 1.019 | 0.991 *** | 0.988 *** | 1.008 | 0.992 * | 0.993 * | 0.999 | 0.991 *** | 0.990** |

ALMON(12)-MQ(20) | 1.101 | 0.990 *** | 0.997 | 1.006 | 0.989 *** | 0.992 ** | 0.987 *** | 0.986 *** | 0.987 *** |

ALMON(20)-MQ(20) | 1.098 | 0.990 *** | 1.000 | 1.006 | 0.989 *** | 0.993 * | 0.987 *** | 0.986 *** | 0.986 *** |

**Table E5.**Relative out-of-sample forecasting precision (the benchmark is the mean squared forecast error of the HAR model in each case).

**Index: DJIA (live)**. Initial series: realized variance (RV). Transformation: $log\left(100\sqrt{252R{V}_{t}}\right)$. Forecasting horizon: one day.

Initial Estimation Sample: | 1–1,000 | 1–1,000 | 1–2,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial Forecast Sample: | 1,001–2,000 | 1,001–3,000 | 2,001–3,000 | ||||||

Type of Forecast: | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling | Fixed | Recursive | Rolling |

HAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

AR(12) | 0.998 | 0.999 | 1.000 | 0.999 | 0.994 | 0.996 | 0.993 | 0.990 | 0.991 |

AR(20) | 1.000 | 1.003 | 1.007 | 0.999 | 0.997 | 1.003 | 0.994 | 0.993 * | 0.998 |

ALMON(12) | 1.015 | 1.007 | 1.001 | 0.998 | 0.994 | 0.993 | 0.984 ** | 0.984 ** | 0.985 ** |

ALMON(20) | 1.011 | 1.005 | 0.999 | 0.996 | 0.993 | 0.992 * | 0.984 ** | 0.984 ** | 0.985 ** |

HAR-MQ(12) | 1.006 | 1.001 | 1.000 | 1.009 | 1.003 | 1.002 | 1.006 | 1.004 | 1.003 |

HAR-MQ(20) | 1.002 | 1.002 | 1.003 | 1.001 | 1.000 | 1.001 | 0.999 | 0.999 | 0.999 |

AR(12)-MQ(12) | 1.003 | 1.002 | 1.003 | 1.009 | 0.997 | 0.999 | 1.000 | 0.994 | 0.995 |

AR(12)-MQ(20) | 0.992 * | 0.999 | 1.002 | 0.995 | 0.993 * | 0.996 | 0.992 | 0.989 * | 0.990 * |

AR(20)-MQ(12) | 1.007 | 1.006 | 1.010 | 1.011 | 1.001 | 1.007 | 1.001 | 0.997 | 1.002 |

AR(20)-MQ(20) | 1.003 | 1.006 | 1.011 | 1.001 | 0.998 | 1.004 | 0.994 | 0.993 | 0.997 |

ALMON(12)-MQ(12) | 0.999 | 0.997 | 0.996 | 0.997 | 0.993 | 0.992 * | 0.994 | 0.990 | 0.990 |

ALMON(20)-MQ(12) | 0.998 | 0.997 | 0.995 | 0.997 | 0.993 | 0.992 * | 0.994 | 0.991 | 0.990 |

ALMON(12)-MQ(20) | 0.992 | 0.995 | 0.994 | 0.987 *** | 0.988 *** | 0.987 *** | 0.983 *** | 0.982 *** | 0.983 *** |

ALMON(20)-MQ(20) | 0.992 | 0.995 | 0.993 | 0.987 *** | 0.988 *** | 0.987 *** | 0.983 *** | 0.982 *** | 0.983 *** |

## Conflicts of Interest

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^{2}An acronym of Regression Error Specification Test.^{3}In Table 1 the following acronyms are used: AR (autoregressive), ARMA (autoregressive moving-average), BL (bilinear), GARCH (generalized autoregressive conditional heteroscedasticity), FARIMA (fractional autoregressive integrated moving-average), STAR (smooth transition AR), ESTAR (exponential STAR), LSTAR (logistic STAR), SETAR (self-exiting threshold AR), MS (Markov switching), RLS-NS (random level shift–non-stationary), RLS-S (random level shift–stationary).^{4}In a few cases, we slightly modify the original parametrization in order to introduce more persistence in the sample autocorrelation function.^{5}See [54]. Data source: http://realized.oxford-man.ox.ac.uk/media/1366/oxfordmanrealizedvolatilityindices.zip (last accessed on 16 November 2014).^{6}The transformation applied is $r{v}_{t}\equiv log\left(100\sqrt{252R{V}_{t}}\right)$, where $R{V}_{t}$ denotes the initial realized variance series.^{8}Ghysels et al. [56] found that this was the case for the S&P 500 realized volatility based on a consideration of the exponential Almon polynomial constraint. Furthermore, the gains in the forecasting performance of models with MQ presented later in this study can also be viewed as additional indirect evidence.^{9}As shown previously, the information criteria selected this order in an unconstrained linear autoregression, as well as the ALMON model.^{10}Whenever the models are nested or overlapping and the asymptotics is used where the out-of-sample forecasting interval relative to the total number of observations is converging to a non-zero fraction, the [64] test is conservative (see, e.g., [65], [66] and [67]). However, whenever we used the Enc-t test statistic (for forecast encompassing) relying on the results of [66], the qualitative picture did not change (available upon request from the authors).^{11}A part of this comes from the fact that we did not align the estimation samples here, but this does not change the conclusion.^{12}Besides the previously defined abbreviation of S&P 500, we will use hereafter the following additional acronyms: FTSE (Financial Times Stock Exchange), DAX (Deutscher Aktienindex), AORD (All Ordinaries), DJIA (Dow Jones Industrial Average).^{13}The linearity was strongly rejected using the heteroscedasticity-consistent MQ test (p-values for at least some MQ windows were not higher than 0.02), whereas the p-values of the RESET tests were highly insignificant for all but the Russell 2000 index.^{14}It is of interest to note that the p-values of the MQ tests as in Remark 3, where either the HAR or the exponential Almon restrictions were imposed on the linear autoregressive part of the AR-MQ model, were slightly worse than that of the test without the constraints. In light of the results presented in Subsection 3.5, this fact can suggest that there is still room for searching for a more adequate restriction of the linear part of the model.

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Ishida, I.; Kvedaras, V. Modeling Autoregressive Processes with Moving-Quantiles-Implied Nonlinearity. *Econometrics* **2015**, *3*, 2-54.
https://doi.org/10.3390/econometrics3010002

**AMA Style**

Ishida I, Kvedaras V. Modeling Autoregressive Processes with Moving-Quantiles-Implied Nonlinearity. *Econometrics*. 2015; 3(1):2-54.
https://doi.org/10.3390/econometrics3010002

**Chicago/Turabian Style**

Ishida, Isao, and Virmantas Kvedaras. 2015. "Modeling Autoregressive Processes with Moving-Quantiles-Implied Nonlinearity" *Econometrics* 3, no. 1: 2-54.
https://doi.org/10.3390/econometrics3010002