Interpreting time series regression coefficients

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The standard ways to do this are either: (1) (easier) test the intercept is equal to zero in the time series regression of excess returns of a security/portfolio/etc on market excess returns or (2 Interpretation. For monthly returns of the S&P 500 index, we demonstrate statistical as well as economic evidence of out-of-sample predictability: relative to an investor using the historic mean, an investor using our methodology could have earned consistently positive utility Chapter 3. Jun 15, 2019 · This tutorial walks through an example of a regression analysis and provides an in-depth explanation of how to interpret the regression coefficients that result from the regression. 8812. Let’s suppose that y, x2, and x3 are continuous variables where y = the annual salary of a person in a company, x2 = number of years the person has worked in the company, and x3 = the Oct 18, 2023 · The coefficient for temperature has now drastically changed (from 11. In other words, the slope of a line is the change in the y variable over the change in the x Oct 18, 2019 · I interpret it as describing the long-run equilibrium of the system. An example illustrates the technique and the awkward properties of regression coefficients in the In the Stata regression shown below, the prediction equation is price = -294. Or too many outliers. Note that I had to multiply by 100 here. Interpreting regression coefficients for log-transformed variables. Here is a description of the method I'm trying to employ. This is called a semi-log estimation. Would I be interpreting the long run and short run coefficients of the UECM in the same way? Sep 24, 2017 · Here is the output in stata after doing one example regression with the Globalisation-Index ("Glob", reaching from 0 to 100) and health expenditures per capita (in $) as regressors. Now when I regress it against the dependent variable the coefficient is significant. In situations when the proportional hazards assumption of the Cox Oct 24, 2017 · Understanding the table of fitted values for the forecast also helps. I have performed a difference-in-differences analysis but I'm not sure how to interpret the results. 2665. For this example, the R estimate of the AR (1) coefficient is: Dec 19, 2020 · Linear Regression: Time Series Analysis Hi everyone! This is a quick and simple Time Series: Linear Regression analysis - Output interpretation in Eviews. In the relationship between two time series ( y t and x t ), the series y t may be related to past lags of the x -series. One defining characteristic of a time For example, suppose a simple regression equation is given by y = 7x - 3, then 7 is the coefficient, x is the predictor and -3 is the constant term. The major complication, in that case, is that the usual estimator for the standard errors of the coefficients becomes Dec 23, 2022 · Yang, J. 4) 1/2. It is calculated as 1 – ll (model) / ll (null) = 0. This is known as a semi-elasticity or a level-log model. We see a regularly repeating pattern that seems to repeat about every 30 or so time periods. Parametrizations of Segmented Regression. If the residual series is unit-root nonstationarity, take the first difference of both the dependent and explanatory variables. arima in R on my monthly time series of the annual change in House prices. 2. Regression coefficients are estimates of the unknown population parameters and describe the relationship between a predictor variable and the response. It also gives an understanding of the basic features of a regression outp Oct 2, 2023 · After the change, a new regression model is fit and the process continues until the end of the time series. Yesterday's Sales (lag 1 variable) is highly significant(p<0. L1 and unemploy. 1955 (mpg) + 1767. Nov 17, 2014 · More erratic series are explained by cycles of lower frequency that are repeated every fewer observations. 42 - telling you that price is predicted to increase 1767. 85). Specify the quarterly date format. 8 = . Jan 29, 2023 · Multicollinearity occurs when independent variables in a regression model are correlated. I have a regression on the form: Y = α + β1 (treatment) + β2 (time) + β3 (treatment∗time) The thing is that neither coefficient is significant but the F-test shows significance on the 0. Then ln 9 − ln 8 = . Feb 19, 2020 · Regression allows you to estimate how a dependent variable changes as the independent variable (s) change. 01-level. Landing time series in Nov 26, 2015 · 25 Nov 2015, 16:41. 1955 when mpg goes up by one, and is predicted to be 11905. In this page, we will discuss how to interpret a regression model when some variables in the model have been log transformed. I'm fitting a segmented (piecewise) linear regression on time-series data with break points representing an intervention. If my preferred specification is to interpret the model as in Pseudo R2 – This is McFadden’s pseudo R-squared. For each row corresponding to time t, the values of level and trend are the smoothed values for time t-1, and the value of season is the smoothed value for t-p. It looks like the errors from Step 1 have an AR (1) structure. But now I am confused on the interpretation of those variables. To say that βi ≠ 0 β i ≠ 0 is to say that Xi X i is associated with Y Y after accounting for the effects of the other X X s if linearity, additivity, and possibly other model assumptions hold. ”) Sep 13, 2020 · 1. This is practically useless, so we exponentiate the parameter and interpret this value as a geometric mean difference. 118 or 11. We have created an interface-based tool (Arévalo et al. So the correlation coefficient is: r = (0. The stimulus was applied for 16 time periods (of 2 seconds) and not applied for another 16 time periods (of 2 seconds). For ease of comparability, column 1 repeats the results from our initial regression further above. Step 3. (about 1/3 ~ 1/4) But the sign and relative size of these coefficients (compared to the coefficients of the other dummies) remains almost the same. To do this, I'm scaling the numbers for the last N months by X%, and using the xreg parameter to change the model from a differenced AR (1) to a regression with time-series errors. As calendar year is the predictor, the intercept here is the value predicted for year 0. Differently, a time series regression model includes a time dimension and can be written, in a simple and general formulation, using just one explanatory variable, as follows: yt =β0 +β1xt +ϵt y t = β 0 + β 1 x t + ϵ t. 03 Volume: 5 (originally measured in Millions) Income: 4 Interest: -. December 1989. 1335 / 9 = 53. yt=443. 8812 = 8. A link to the chapter is given below: . The residual mean squares is calculated by residual SS / residual df. The formula for Regression Coefficients in Simple Regression Models is: b = n Σ X Y – Σ X Σ Y n Σ X 2 – ( Σ X) 2. In Cornell Statistical Consulting Unit Statnews; CSCU: Ithaca, NY, USA, 2012; Volume 83, p. Structural Vector Autoregressive Modeling and Impulse Responses pp. If you log transform an outcome and model it in a linear regression using the following formula specification: log(y) ~ x, the coefficient is a mean difference of the comparing adjacent units of X X. You survey 500 people whose incomes range from 15k to 75k and ask them to rank their happiness on a scale from 1 to May 5, 2021 · Suppose we have a time-series regression given by: Yt = β0 +β1Yt−1 Y t = β 0 + β 1 Y t − 1. 42Qtr2+78. I am using SAS to run an ARIMAX model on two time series where the response series= asthma hospitalizations and the explanatory/input series= influenza hospitalizations. But the meaning is neither interesting nor useful. The coefficients are significant and have the expected signs assumed by theory. ”. After much researching I the following reference was the most useful to me when trying to interpret the findings of a vecm: Helmut Lütkepohl, Markus Krätzig. The linear regression model assumes there is a linear relationship between the forecast variable and the predictor variables. csv format). The correlation coefficient will be positive because both the coefficients are positive. 001) we can reject the null hypotheses of the coefficients being not significant. It is a parameter estimate describing the relationship between one of the independent variables in your model and the dependent variable. Based on this coefficient, theoretically, it indicates the quality of the regression model. One difference from standard linear regression is that the data are not necessarily independent and not necessarily identically distributed. Is it just so simple that there is no need to state it formally? I'm working on a project on time series regression. 6 and 0. May 1, 1978 · Interpreting Polynomial Regression. Jan 8, 2021 at 15:13. To complete the answer of Glen_b and his/her example on random walks, if you really want to use Pearson correlation on this kind of time series (St)1≤t≤T ( S t) 1 ≤ t ≤ T, you should first differentiate them, then work out the correlation coefficient on the increments ( Xt = St −St−1 X t = S t − S t − 1) which are (in the 8. However I'm not sure how to interpret this. First, let STATA know you are using time series data. The parameters in the model are: B 2, the second regression coefficient. I have 47 columns I'm calculating the seasonal index for I am trying to understand the coefficients retrieved from running auto. In this example, regression MS = 546. Sep 5, 2023 · The coefficient of determination takes values between 0 and 1. I am doing a time series regression between 2 variables. This brief hands-on tutorial shows how to interpret a regression output in Stata13. Following Box-Steffensmeier & Jones (2004) I included interactions with time for these covariates. (Or, you could consider log transformation. For this it is essential to organize the data in a counting process style. Again, this variance inflation factor tells us that the variance of the weight coefficient is inflated by a factor of In a regression model, the partial effect or marginal effect of a regression variable is the change in the value of the response variable for every unit change in the regression variable. 5% increase. Are log difference time series models better than growth rates? 4. Poisson regression does not have an equivalent to the R-squared found in OLS regression; however, many have tried to derive an equivalent measure. To explain the two common parametrizations of segmented regression for ITS, we consider the setting of a single interrupted time series collected from one unit (for example, a single clinic) with a continuous outcome variable. Feb 17, 2012 · The simple regression model is. In statistics, AIC is used to compare different possible models and determine which one is the best fit for the data. Admittedly, I have only little knowledge of the concepts behind it. If on the other hand, you had [Income at date] = B0 + B1 [Region at date-1] + B2 [Age at date-1] + B3 [Income at date-1] where we have added a third predictor which is the same as the dependent variable but lagged, then we are looking at time series modeling, more specifically autoregressive May 28, 2020 · I'm working on extracting the trend/seasonal relatives of my TS data. If a constant were to be in the differenced equation, that would correspond to a linear trend in the levels equation. For example, yt = 9 y t = 9 and yt−1 = 8 y t − 1 = 8. The regression mean squares is calculated by regression SS / regression df. In seasonal time series, it is common to see cycles related to seasonal frequencies that are repeated every 2, 6 or 12 moths. – Richard Hardy. 8 to 0. There are a variety of pseudo-R-square statistics. Mar 18, 2019 · How to interpret log-log regression coefficients for other than 1 or 10 percent change? 2 Interpretation of coefficient in log-linear model with share predictor Apr 6, 2017 · Hereby, I basically followed the advice of Tsay (2010, 3rd edition, p. Firstly, time variables themselves: trend over time, seasonality, and business cycles, are often useful in describing and/or predicting the behavior of a time series. Introduction. And the correlation coefficient is the geometric mean of both the coefficients. Interpreting Regression Coefficients. Simple linear regression example. The data exhibit strong seasonality with a trend. $\begingroup$ Thank you for your answers, you make things clearer, but I still haven't understood completely what is the interpretation of the coefficients when all of my variables are scaled, and when only the predictors are scaled. In: Applied time-series economics. 2−59. Such variable can be analyzed with the Cox regression model to estimate its effect on survival time. A standard regression model Y Y = β β + βx β x + ϵ ϵ has no time component. Could you please point out where I am getting it wrong: 1) The R squared seems very low -- this indicates a weak linear relationship. There are four key assumptions that multiple linear regression makes about the data: 1. It measures by how much postgraduate education raises income on average. The basic problem we’re considering is the description and modeling of the relationship between two time series. 87Qtr3 Interpret the regression coefficient for the Qtr2 variable. 12% or, in decimal form, 0. As time increases by one quarter, revenue decreases by $59. The model looks good, with four of the six regressors significant: Here are the OLS residuals: I think that you need to remember that ARIMA models are atheoretic models, so the usual approach to interpreting estimated regression coefficients does not really carry over to ARIMA modelling. My goal is to assess wether the intervention significantly changed the trend (slope) of the the regression line. I have gone through and assessed goodness of fit, checked residuals, checked ccf, etc and I'm finally at my final model with all the estimates. It wouldn't make much sense to change my interpretation now to a unit increase in temperature resulting in an additional 0. Sep 3, 2015 · There are few explanations I can find that describe how to interpret linear regression coefficients after differencing a time series (to eliminate a unit root). Although the metrics and coefficients resemble those from linear regression, I was somehow Jun 1, 2016 · Note that the order of integ ration of each series in a regression model is determined by the unit root test. Specifying the correct model is an iterative process where you fit a model, check the results, and possibly modify it. The example data can be downloaded here (the file is in . In the Data Analysis popup, choose Regression, and then follow the steps below. The Akaike information criterion (AIC) is a mathematical method for evaluating how well a model fits the data it was generated from. You are a social researcher interested in the relationship between income and happiness. I don't understand how to interpret the coefficient from a Poisson regression relative to the coefficient from an OLS regression. 118 ln. I use regression to model the bone By Jim Frost. In order to interpret (or understand) estimated ARIMA models, one would do well to be cognizant of the different features displayed by a number of 9. 2. The variables in the model are: e, the residual error, which is an unmeasured variable. Is. The same is true if the lag is changed to two months. To the extent that you cannot directly interpret the coefficient, the same likely holds for its sign as well. In the simple linear regression below, the quantity 0. , the sum over all squared differences between the observed values of the i th observation of y i and the corresponding predicted values $ {\hat{y}}_i $) is minimized. 2020 ) that facilitates the exploration of time series of Landsat observations and the CCDC results. – Christoph Hanck. Table of the fitted OLSR model’s regression coefficients from fitted the model 100 times, each time on a randomly selected sample of 50 vehicles (Image by Author) Each row in this table corresponds to the fitted values of the regression coefficients obtained from fitting the OLSR model on a randomly selected sample of 50 vehicles. g. 1007/BF02685335. Independence: The residuals are independent. In this equation, +3 is the 8. Lehigh University. A time series plot follows. r = (0. 8%, which is the logarithmic approximation to the actual 12. You begin by creating a line chart of the time series. I used the dynlm library in R. The paper explains in fairly simple terms the theory Dec 21, 2023 · To economize on space, we summarize the results for the speed-of-adjustment coefficient (ADJ) and the long-run coefficients (LR, excluding the linear time trend) in table 1 instead of showing detailed Stata output. 68151. Set aside the fact that there was no year 0 and years are reckoned in retrospect, in the calendar you are using, to have run Feb 21, 2022 · If you are just looking for an interpretation of the output of such regression, it seems that the coefficients for both unemploy. DOI: 10. Feb 18, 2016 · 1. One might also say that a nonzero coefficient implies that the The equation is estimated by converting the Y values to logarithms and using OLS techniques to estimate the coefficient of the X variable, b. , the dependent variable abbreviated as DV) given a one-unit change in May 4, 2016 · 2. When doing so, I obtain the following outcome: When doing so, I obtain the following outcome: May 24, 2023 · In the case of a level regression (no log transformation), with the regression coefficients corresponding to a partial derivative (∂ Y / ∂ X ), a change of one unit in X implies a change of β₁ unit in Y (with Y the dependent variable, X the independent variable, and β₁ the regression coefficient associated with X). Conversely, smaller coefficient of determination values, even approaching 0, suggest a less effective Dec 1, 1989 · Interpreting semilogarithmic regression coefficients in labor research. This is given by the p-values for such coefficients: being "small" (less than 0. I didn't know that there was something else (or such a big difference between "normal" polynomials and orthogonal polynomials), and the examples I saw online all used poly() for fitting, especially with ggplot – so why wouldn't I just use that A univariate time series is a sequence of measurements of the same variable collected over time. The variables in the data set are writing, reading, and math scores ( write, read and math ), the log transformed writing ( lgwrite) and Mar 20, 2019 · Mean Squares. For small s m a l l changes, you can interpret logged differences as percentage changes after multiplying by 100. The CAPM says that expected returns of any tradable security are linear in its market beta (i. Therefore, the variance inflation factor for the estimated coefficient Weight is by definition: VIFWeight = Var(bWeight) Var(bWeight)min = 1 1 −R2 Weight = 1 1 − 0. 4. Jan 13, 2015 · 18. After pondering over questions and answers on SO and other sources it is not clear as to what value is suitable for First, let STATA know you are using time series data. 85 ice cream sales. 53308 / 2 = 273. If you think that the relation between your target variable and a feature is possibly non-linear, you can add quadratic terms. The regression of time series is similar to other types of regression with two important differences. If the degree of correlation between variables is high enough, it can cause problems when you fit the model and interpret the results. e. I have a time series data set with about 2million observations and 31 variables, which I break to a few thousand using threshold value for my dependent variable. Suppose you have the following regression equation: y = 3X + 5. The independent variable was non stationary so I took first differences to stationarize it. For arguments sake, lets say the parameter estimates are as follows: Rent: 100 Unemployment: -. Le $\begingroup$ @Glen_b Yeah, well, I did type in ?poly to understand the syntax. This correlation is a problem because independent variables should be independent. 42 thousand on average. This may not be surprising. , days, months, years…) passed from the start of the observational period; Oct 1, 2012 · We evaluate predictive regressions that explicitly consider the time-variation of coefficients in a comprehensive Bayesian framework. If the series is stationary at level after performing unit root t est, then it is I (0); As discussed in the example Time Series Regression VI: Residual Diagnostics, autocorrelated innovations in the absence of other CLM violations produce unbiased, if potentially high variance, OLS estimates of model coefficients. AIC is calculated from: Jun 29, 2020 · For the math people (I will be using sklearn’s built-in “load_boston” housing dataset for both models. Suppose I have time series data, my left-hand side variable is number of games won per year, and my main right-hand side variable is NASDAQ value. For an example that includes some of the calculus behind this interpretation you may see this post. Dec 5, 2017 · Time-varying covariance occurs when a covariate changes over time during the follow-up period. A coefficient of determination approaching 1 signifies a better regression model. These are added to give the estimated true level for time t, Xhat. 05 Nov 29, 2020 · Now we interpret the coefficient as a % increase in X, results in a (b/100)*unit increase in Y. Once we have the estimates for the slope and intercept, we need to interpret them. 6 * 0. Is the interpretation that I have given above correct? $\endgroup$ – Jan 30, 2015 · How would I interpret the short run and log run coefficients? For instance, the $\beta$ coefficients on a cross sectional OLS provide information on how a unit change in the independent variable changes the dependent variable. 42. In mathematical terms, it means that the time series equation includes four key coefficients: Y = b0 + b1T + b2D + b3P + e (1. Note the absence of a constant term in the differenced equation: the constant cancels out. Mar 26, 2014 · Statistics using SPSS: The intercept has a meaning here, as in any regression. In our example, this would mean that a 1% increase in years of experience results in a £(b/100) increase in wage. I am using lasso regression in R to do some variable selection. Suppose we have a regression equation of the form (see Multiple Regression Basic Concepts ): y = b0 + b1x1 + b2x2 + b3x3. Time Series Regression. Oct 23, 2019 · When I worked on my first-time series project I struggled with interpreting the results of my model. 1) Where: Y is the outcome variable; T is a continuous variable which indicates the time (e. Dec 20, 2021 · A linear regression model with two predictor variables results in the following equation: Y i = B 0 + B 1 *X 1i + B 2 *X 2i + e i. Authors: Robert j Thornton. generate time=q(1959q1)+_n-1; format time %tq; sort time; tsset time; _n is the observation no. That is, if globalisation increases, poverty is expected to decrease. Journal of Labor Research 10 (4):443-447. That's why people typically look at impulse responses, which are easier to interpret. The vector I'm using as the regressor looks like [0, 0, , 0, 1, 1, 1], where 1s represent the months with the intervention in effect. 0001), we can conclude that there is a significant linear relationship between Removal and OD . Again, differentiating both sides of the equation allows us to develop the interpretation of the X coefficient b: Mar 26, 2020 · Revised on June 22, 2023. In a linear model that contains only linear terms, i. e is stationary as long as e is I(0) or I(1) The differenced equation has no “history. Mar 1, 2016 · How to interpret log-log regression coefficients for other than 1 or 10 percent change? 23. So its obvious that this indicates Examine the AR structure of the residuals. I'm working with monthly data beginning Jan 2016 and ends on Apr 2020. yt = Xtβ+ϵt y t = X t β + ϵ t. Oct 27, 2020 · Assumptions of Multiple Linear Regression. 2) Normal QQ plot shows that there are a Aug 24, 2020 · When one talks about modeling a time series regression, the first thought that comes in one’s “programmed” mind is: - Let’s ensure that stationarity exists for the dependent and independent variables and develop a regression model to capture the short-term trend between the variables. May 16, 2024 · Solution: The two regression coefficients between x and y are 0. responds to change in X 2 after adjusting for simultaneous linear change in the other Nov 11, 2022 · Interrupted time series (ITS) analysis is a time series regression model that aims to evaluate the effect of an intervention on an outcome of interest. Related: How to Read and Interpret an Entire Regression Table. no quadratic, log, and other kinds of nonlinear terms, the main effect of each regression variable is For example, we could specify that the factor is any time series such as the number of tins of beans sold in your local supermarket. May 1978 describe" a curve with a series of linear slopes. In general, the regression coefficient on a dummy variable gives us the average increase in observed when the dummy is equal to 1 (with respect to the base case in which the dummy I am building a regression model of time series data in R, where my primary interest is the coefficients of the independent variables. 0000001) and the coefficient of this lagged variable is 1 Answer. Per the standard interpretation, if β1 = 0. 292 (foreign) + 11905. For linear regression, the target variable is the median value (in $10,000) of owner-occupied homes in a given neighborhood; for logistic regression, I split up the y variable into two categories, with median values over $21k labelled “1” and median values under $21k labelled “0. higher market beta, higher expected returns). 5, which sits in front of the variable Nov 3, 2020 · Download the Excel file that contains the data for this example: MultipleRegression. a = Y ¯ – b X ¯. 51Qtr1−59. Following are the ACF and PACF of the residuals. The sample cross correlation function (CCF) is In the following time series linear regression equation, t is time in quarters and y is revenue in thousands of dollars. Jul 3, 2018 · The coefficients are much smaller when I put Yesterday's Sales to the model. 4 days ago · Or, stated differently, the p-value is used to test the hypothesis that the true slope coefficient is zero. For the cleaning example, we fit a model for Removal versus OD . How would this interpretation change if Y Y is defined as the log of the sum of two other May 9, 2016 · 1) Adding quadratic terms allows for non-linearity (in a linear model). This implies that the errors must have mean The series is actually the average of this process for five different subjects. 0536. Jan 8, 2021 · 1. I fitted a Cox PH Model and upon examination of the Schoenfeld residuals it became apparent that the proportionality assumption was violated by several variables. The basic or unstandardized regression coefficient is interpreted as the predicted change in Y (i. Linear relationship: There exists a linear relationship between the independent variable, x, and the dependent variable, y. 159-196. In linear regression, coefficients are the values that multiply the predictor values. Most often, the measurements are made at regular time intervals. Jan 30, 2024 · So interpret regression coefficients whether or not they are “significant”. In this example, residual MS = 483. In the previous example, is the regression coefficient of the dummy variable. So this command creates a new variable time that has a special quarterly date format. 2 - Interpreting the Coefficients. L12 are significant. 24) 1/2. I'm trying to understand how to interpret the results. If the slope is denoted as \ (m\), then. 292 when the foreign variable goes up by one, decrease by 294. ITS analysis is a quasi-experimental study design instrumental in situations where natural experiments occur, gaining popularity, particularly due to the Covid-19 pandemic. A time series regression forecasts a time series as a linear relationship with the independent variables. (3, 7) The key features of the model equation are a variable for continuous time, a binary indicator denoting the presence of an May 13, 2023 · A regression coefficient is the quantity that sits in front of an independent variable in your regression equation. The details of the original algorithm are described in Zhu and Woodcock ( 2014 ). For example, suppose you're a portfolio manager and you have R2Weight is 88. Feb 20, 2017 · Hoaglin argues that the correct interp retation of a regression coefficient is that it “tells us how Y. Because our p -value is very small (<. 1 increase in Yt Y t. 2 Cross Correlation Functions and Lagged Regressions. Suppose the equation of the best-fitted line is given by Y = aX + b then, the regression coefficients formula is given as follows: a = n(∑xy)−(∑x)(∑y) n(∑x2)−(∑x)2 n ( ∑ x y) − Mar 26, 2016 · To estimate a time series regression model, a trend must be estimated. In Excel, click Data Analysis on the Data tab, as shown above. ) 2) Significance of quadratic terms could signal that the relation is non-linear. Estimate the AR coefficients (and make sure that the AR model actually fits the residuals). The line chart shows how a variable changes over time; it can be used to inspect the characteristics of the data, in particular, to see whether a trend exists. In that case, the second regression more clearly converts any factor exposures ($\beta_{i,SMB}$, etc. However, I'm struggling to explain why this is the case, since mathematically: $$\gamma_1 = \frac{\partial C_t}{\partial Y_t}$$ which is exactly the definition of the MPC. Dec 3, 2021 · The aim of linear regression analysis is to estimate the coefficients of the regression equation b 0 and b k (k∈K) so that the sum of the squared residuals (i. Recall from the beginning of the Lesson what the slope of a line means algebraically. ) into an investable strategy that would earn the risk premium for that factor in the marketplace. 42 when both mpg and foreign are zero. 1 β 1 = 0. 1 for example, a 1 unit increase in Yt−1 Y t − 1 is associated with a 0. 96): Financial Time Series: Fit the linear regression model and check serial correlations of the residuals. nt nm gi wa et hk vc bp to oq