Leveraging Reverse Regressions for Bias Diagnosis in the Digital Economy Datasets

Volume : 6 Issue : 2 Year : 2025 Pages : 47-57 e-ISSN : 2717-9230 47 LEVERAGING REVERSE REGRESSIONS FOR BIAS DIAGNOSIS IN THE DIGITAL ECONOMY DATASETS Richard MULENGA a a ZCAS University, Department of Economics, Lusaka, Zambia, [email protected], hCps://orcid.org/0009-0003-00657432. ABSTRACT: This paper evaluates reverse regression in simula2ons and applica2ons mo2vated by the digital economy data. Data from digital pla:orms ranging from e-commerce transac2ons to user-generated content offers vast poten2al for economic analysis, yet it frequently suffers from measurement errors and endogeneity problems. With digital pla:orms producing vast amounts of data that are frequently user-created, collected, or compiled, researchers encounter growing difficul2es in valida2ng data reliability. The reverse regression provides a unique diagnos2c tool set for iden2fying and correc2ng biases when the standard assump2ons of Ordinary Least Squares (OLS) are not sa2sfied. This is par2cularly true in contexts like gig work income reports, online adver2sing, and consumer trends inferred from internet ac2vi2es. Based on the simulated digital data of a medium enterprise business digital sales data associated with adver2sing expenditure reported via Google or Meta dashboards, this study finds that the forward regressions are biased or aMenuated. The study therefore recommends that reverse regression involving the digital pla:orm data be applied as a diagnos2c and correc2ve tool set in early-stage econometric diagnos2cs, especially when robust instrumental variables are unavailable. Keywords: Reverse regressions, Digital Pla9orm, Digital Economy, Measurement Error, Meta Dashboards. RECEIVED: 17 June 2025 ACCEPTED: 20 November 2025 DOI: hGps://doi.org/10.5281/zenodo.18056973 CITE Mulenga, R., (2025). Leveraging Reverse Regressions for Bias Diagnosis in the Digital Economy Datasets. European Journal of Digital Economy Research, 6(2), 47-57. hGps://doi.org/10.5281/zenodo.18056973 Research Paper Mulenga 48 1. INTRODUCTION The digital economy has changed how data is produced, gathered, and analyzed. Data from digital platforms ranging from e-commerce transactions to user-generated content offers vast potential for economic analysis, yet it frequently suffers from measurement errors and endogeneity problems. The proliferation of digital platforms like Uber and Airbnb has transformed markets, consumer habits, and economic frameworks. These platforms produce extensive data that researchers and policymakers utilize to comprehend market trends, consumer choices, and employment results (Goldfarb & Tucker, 2019). However, data from platforms frequently experience measurement inaccuracies resulting from self-reported information, algorithmic changes, and insufficient validation. Moreover, endogeneity problems often occur from simultaneity and missing variables, undermining causal inference. This study evaluates the capacity of reverse regression as a diagnostic tool to address these econometric issues in the context of the proliferation of digital economy data sets1. Reverse regression involves regressing the independent variable of interest against the dependent variable, assessing whether the assumed direction of causality is maintained upon examination (Hansen, 2015; Phillips & Shi, 2018). Initially created for detecting measurement errors (See, for example, Hausman, 2001; Cochrane, 2001) reverse regression has seen restricted use in studies related to the digital economy. This research seeks to fill that gap by investigating the application of reverse regression for identifying measurement error and endogeneity within platform-based datasets. We use actual data from Uber and Airbnb to demonstrate the practical uses and consequences of this diagnostic method. 1.1. Digital PlaLorm-Based Data and Econometric Challenges Digital pla9orm-based datasets, such as those derived from Uber, Airbnb, facebook, Amazon, Spobfy and Google, provide innovabve insights but 1 There are various definiQons of digital economy offered in the literature. This study adopts the OrganizaQon for Economic CooperaQon and Development, OECD (2020) and the United NaQons Conference on Trade and Development, UNCTAD (2019) definiQons. Digital Economy encompasses "all are not originally collected for scholarly research, which presents disbnct challenges (Einav & Levin, 2014). Digital pla9orms enable users to connect, communicate and engage with each other. Addibonally, digital pla9orms provide infrastructure for buying, selling and exchanging goods and services (UNCTAD, 2019). Measurement inaccuracies can arise from pla9orm design elements like rabng systems, user behaviors such as self-selecbon, or data processing techniques including search ranking algorithms. Addibonally, endogeneity issues are widespread in digital pla9orm-based data due to the nature of two-sided markets and the presence of feedback loops among variables (Bajari et al., 2015). Measurement errors are common in digital pla9orm data, that oden contain variables subject to noise. For instance, rabngs can be affected by subjecbve biases and social desirability effects (Luca, 2016; Nosko & Tadelis, 2015). Locabon informabon may lack precision due to reliance on Internet Protocols (IP) based geolocabon techniques (Chen et al., 2016). Furthermore, prices are frequently subject to dynamic and algorithmic adjustments, creabng ambiguity regarding the actual observed values (Einav & Levin, 2014). Such inaccuracies diminish the reliability of key regressors, availability of robust instrumental variables and can result in biased coefficient esbmates in empirical forward (direct) based regression analyses (Bound, Brown, & Mathiowetz, 2001). In digital marketplaces, simultaneity or endogeneity problems are frequently observed, whereby prices and demand levels influence each other in real bme (Athey & Imbens, 2017). Addibonally, pla9orm visibility factors, such as search rankings, are impacted and influenced by click-through rates (Ghose & Yang, 2009). Furthermore, unobserved variables such as changes in pla9orm policies or consumer expectabons can simultaneously affect both the explanatory variables and the outcomes under considerabon (Bajari et al., 2015). economic acQvity that is supported by informaQon and communicaQons technologies (ICT), including e-commerce and digitally delivered services" (OECD, 2020). Digital economy entails "the global network of economic and social acQviQes that are enabled by ICT, including e-commerce, digital pla[orms, and data-driven services" (UNCTAD, 2019). Leveraging Reverse Regressions for Bias Diagnosis in the Digital Economy Datasets 49 Although recent developments in causal inference methods, including instrumental variables (IV) and difference-in-differences (DiD), have improved analybcal approaches, many studies sbll face difficulbes in establishing credible idenbficabon within digital data pla9orms. The review of extant literature seems to indicate that reverse regression remains an underublised tool as a diagnosbc and correcbve tool in the early stages of digital data analyses, and as a correcbve tool for bias in financial and economic studies. Addibonally, studies employing reverse regressions focusing on the digital economy are scarce. Therefore, this study fills this gap by examining the applicabon of reverse regression as a diagnosbc methodology for idenbfying measurement errors and endogeneity within econometric models, with parbcular emphasis on the context of the digital economy empirical data analyses (Athey & Imbens, 2017; Goldfarb & Tucker, 2019). 2. LITERATURE REVIEW 2.1. Empirical Literature Review The concept of using reverse regression dates back to Cochrane (2001) and Hausman (2001), who asserted that assessing how future variables relate to current ones can help in understanding equilibrium relabonships and the sources of predictability. Reverse regression has aGracted scholarly interest in econometrics due to its effecbveness in idenbfying classical measurement error, which occurs when an explanatory variable is observed with noise (Hausman, 2001). In tradibonal contexts, the bias caused by aGenuabon in ordinary least squares (OLS) esbmates can be detected by comparing the coefficients obtained from direct and reverse regressions. A notable discrepancy between these coefficients indicates the likelihood of measurement error (Griliches, 1986). Addibonally, reverse regression has been employed to idenbfy simultaneity bias in models where the causal relabonship is uncertain (Angrist & Pischke, 2009). For instance, in supply and demand frameworks, reverse regression aids in disbnguishing whether price influences quanbty or if the causality runs in the opposite direcbon. Dufour and Kang (2022) re-evaluate the concept of reverse regression (RR) within the framework of the classical linear regression (CLR) model, emphasizing distribubonal symmetry and its consequences for hypothesis tesbng. Although reverse regression has historically been regarded as a special case, oden associated with measurement error correcbon strategies. This study aimed to formalize the RR theorebcal foundabon and demonstrate its pracbcal and inferenbal significance. The authors establish a rigorous stabsbcal framework that characterises reverse regression as a mirror counterpart to forward regression, assuming a jointly Gaussian mulbvariate distribubon. Their work advances the literature on reverse regression by providing new theorebcal insights. Through a thorough mathemabcal analysis, Dufour and Kang (2022) reveal profound distribubonal symmetries connecbng reverse and forward regression under standard linear assumpbons. Overall, the research affirms reverse regression’s role as a valuable addibon to the stabsbcian’s early-stage econometric analysis toolkit. Building on this idea, Phillips and Shi (2018) contribute to a growing field of research in econometrics and financial economics focused on idenbfying, understanding, and dabng speculabve financial bubbles. They introduce reverse regression as a new method to detect and date stamp explosive financial bubbles. Instead of predicbng future prices based on current data, they suggest esbmabng the explosive behavior by regressing current prices backward onto future prices. Authors assert that RR esbmates seem to be less impacted by measurement errors. Wei and Wright (2013) add to the research on long-term forecasbng, especially when it comes to predicbng asset returns and macroeconomic indicators using reverse regression methods. These long-horizon regressions, which involve forecasbng a variable like stock returns or GDP over several future periods, oden face challenges such as small-sample bias, overlapping data points, and poor performance when tested out-ofsample. Wei and Wright invesbgate whether flipping the regression predicbng future values based on current variables can lead to beGer stabsbcal properbes or new insights. Cready, HurG, and Seida (2000) examine a persistent empirical issue in the fields of accounbng and finance: the esbmabon of the relabonship between stock returns and accounbng earnings. Convenbonal approaches typically involve forward regressions, where stock returns are regressed on reported earnings or earnings changes to assess their influence on market valuabon. Nonetheless, these regressions are suscepbble to measurement error, parbcularly Mulenga 50 when the explanatory variable, earnings in this context, is subject to noise or imperfect measurement. The authors invesbgate whether employing reverse regression can mibgate these measurement errors and produce more accurate inferences regarding the pricing of earnings informabon. They argue that, according to econometric theory (Hausman, 1978; Griliches & Ringstad, 1970), reverse regression can, under specific assumpbons, provide unbiased esbmates of the relabonship between the variables, especially when the dependent variable (earnings) is measured with less error than the independent variable (returns). Overall, the study asserts that reverse regressions serve as a diagnosbc and correcbve tool for addressing measurement errors in empirical early-stage diagnosbc test analyses. Goldberger’s (1984) publicabon serves as a fundamental cribque and clarificabon regarding the use of reverse regression within econometric analysis. Goldberger (1984) provides a caubonary perspecbve on the inappropriate use of reverse regression. Although it does not enbrely prohibit its applicabon, the paper advocates for its use in suitable contexts such as diagnosbc procedures, exploratory data analysis, or assessments of symmetry, rather than in straigh9orward structural esbmabon. The author caubons that any rigorous applicabon of reverse regression in contemporary research must consider the insights and limitabons highlighted by this influenbal work. Greene (1984) invesbgated reverse regression methodologies within the framework of wage discriminabon analysis, with a focus on labor economics. Greene’s (1984) study offers an algebraic and stabsbcal analysis of the consequences of interchanging the roles of dependent and independent variables in discriminabon research. He demonstrates that reverse regressions are not merely symmetrical counterparts to forward regressions; the esbmated coefficients vary due to differences in variance, covariance structures, and group sizes. The paper provides explicit algebraic derivabons illustrabng that reverse regression yields esbmates of discriminabon that are both stabsbcally and economically disbnct from those obtained through forward regression, rather than being simple reexpressions of the same underlying relabonship. The author explicitly warns against interprebng reverse regression as a dependable approach for measuring discriminabon, parbcularly in the absence of strong assumpbons regarding homoscedasbcity, normality, and linearity. Racine and Rilstone (1995) reevaluate the issue of reverse regression in light of cribcisms raised by Goldberger (1984) and Greene (1984). The phenomenon known as the "reverse regression paradox" describes the oden unexpected and asymmetric outcomes that emerge when the roles of dependent and independent variables are interchanged in a regression analysis. This paradox has generated confusion, parbcularly within disciplines such as labor economics and finance, where reverse regression techniques have been employed both for diagnosbc purposes and inferenbal analysis. Schaefer and Visser (2003) contribute to the ongoing scholarly discourse on reverse regression within the framework of employment discriminabon analysis, with parbcular emphasis on wage and salary differenbals. Their research serves as a pracbcal addibon to the fields of applied econometrics and forensic economics, examining how various regression techniques— namely forward regression, reverse regression, and orthogonal regression produce divergent conclusions regarding the presence and extent of discriminabon. The authors contend that orthogonal regression provides a balanced approach by mibgabng the arbitrary asymmetry inherent in forward and reverse regression methods. Cready et al. (2000) examine the applicabon of reverse regression (RR) within the context of financial accounbng, with a specific focus on earnings–returns analyses. While convenbonal approaches typically model stock returns as a dependent variable influenced by earnings through forward regression, RR offers an alternabve by treabng earnings as a funcbon of returns, cibng advantages such as improved interpretability and mibgabon of aGenuabon bias. The researchers ublize both simulated datasets and real financial data to assess and compare the stabsbcal characterisbcs and interpretabve implicabons of forward versus reverse regression models. Addibonally, they invesbgate how the esbmated RR coefficients are affected by various sample selecbon criteria, definibons of earnings, and different bme horizons, aiming to determine whether RR provides more consistent or comprehensible insights. The results reveal that reverse regression yields substanbally different coefficient esbmates compared to forward Leveraging Reverse Regressions for Bias Diagnosis in the Digital Economy Datasets 51 regression, especially in cases where earnings are noisy or contain transient components. Dereziński and Warmuth (2018) introduce reverse iterabve volume sampling (RIVS), an innovabve approach for selecbng subsets in linear regression analysis. This method intersects the fields of linear regression, randomized algorithms, and subset selecbon techniques commonly employed in large-scale data applicabons and kernel methods. It builds upon tradibonal volume sampling techniques by implemenbng a reverse, or backward, strategy that enhances both computabonal efficiency and stabsbcal robustness. Their work extends exisbng literature by developing a reverse iterabve volume sampling—that circumvents the need to compute large determinants, thereby increasing scalability. Dereziński and Warmuth (2018) present a stabsbcally rigorous and computabonally efficient methodology for subset selecbon in linear regression, transforming volume sampling into a reverse eliminabon process that preserves its desirable stabsbcal qualibes while facilitabng applicabon to large datasets. Zeng et al. (2008) examine the issue of univariate calibrabon in the context of heteroscedasbcity. The authors revisit the concept of reversed regression (RR), where concentrabon is regressed on signal rather than the tradibonal approach of regressing signal on concentrabon, and assess whether RR offers advantages under heteroscedasbc condibons, parbcularly in the calibrabon of instruments. Their research extends prior stabsbcal and chemometric studies by concentrabng on heteroscedasbc data, a common yet frequently overlooked characterisbc in pracbcal calibrabon scenarios. The findings indicate that RR produces less biased esbmates of concentrabon, especially when the signal is affected by substanbal and non-uniform measurement errors, characterisbc of heteroscedasbcity. Their case study supports the broader conclusion that reverse regression can be stabsbcally advantageous when the objecbve is to infer an unobserved input variable from noisy output data. Otero and Baum (2018) examine the robustness and efficacy of unit root tests, specifically the Dickey–Fuller (DF) and augmented Dickey–Fuller (ADF) procedures, by proposing a framework that incorporates both forward and reverse regression approaches. Their goal is to improve the stabsbcal power and dependability of tradibonal unit root tesbng methods by analyzing how the direcbonality of the regression influences inference, parbcularly in small sample contexts or processes that are close to a unit root. Their results suggest that forward and reverse ADF tests exhibit comparable power under the null hypothesis, yet their performance may differ under certain alternabve hypotheses, such as processes nearing a unit root. Researchers and pracbboners can leverage these combined forward-reverse ADF tesbng strategies to enhance diagnosbc accuracy of unit root conclusions, especially when working with limited data or data suscepbble to measurement errors. Ash (2014) cribcally analyzes the applicabon and interprebve challenges associated with reverse regression techniques within the legal fraternity, with a parbcular focus on employment discriminabon cases. The chapter scrubnizes the growing reliance of defendants on reverse regression (RR) as an evidenbary strategy to undermine discriminabon claims, highlighbng the tendency for such methods to depend on flawed or inappropriate logical assumpbons. Ash (2014) emphasizes important methodological concerns, notably that reverse regressions may violate fundamental linear regression assumpbons such as exogeneity and proper model specificabon. The study warns that courts unfamiliar with the stabsbcal limitabons of RR may misinterpret its results, leading to potenbal misjudgments in legal proceedings. Wei and Wright (2009) examine the methodology for construcbng confidence intervals within longhorizon predicbve regressions, with parbcular emphasis on applicabons in financial economics where the predictability of macroeconomic indicators is evaluated using variables such as dividend-price rabos, interest rates, or inflabon rates. The study is driven by the recognized challenge that convenbonal confidence intervals for long-term coefficients frequently prove to be misleading primarily due to issues such as persistent regressors, small sample biases, and autocorrelabon in error terms. To address these concerns, the authors introduce the RR approach. Authors assert that the sampling distribubon of the RR-based esbmator is less affected by the persistence of predictors and that RR facilitates more precise finite-sample inference compared to tradibonal forward regression techniques. Cartwright and Riabko (2024) examine the impact of temporal aggregabon, specifically, the transibon Mulenga 52 from high-frequency data such as daily observabons to lower-frequency levels like weekly or monthly, on the accuracy of parameter esbmabon and the validity of inference within reverse regression models applied to commodity markets. This chapter analyses how such aggregabon influences esbmabon outcomes, revealing that while it may enhance R² values, it also tends to inflate standard errors and diminish the reliability of inferenbal conclusions, ulbmately resulbng in less accurate forecasts. The research highlights the importance of exercising caubon when employing aggregated data for reverse regression analyses in financial contexts. Chen (2011) invesbgates the performance of forward (direct) and reverse regressions in esbmabon, employing both ordinary least squares (OLS) and instrumental variables (IV) methodologies. The research focuses on analyzing returns to scale and technological progress within the U.S. manufacturing sector over a span of approximately fidy years. Operabng within an error-in-variables (EIV) framework where both input and output growth rates are subject to noise measurement, the study demonstrates that OLS esbmates, whether direct or reverse, are inconsistent, with the reverse OLS generally exhibibng greater precision under the assumpbon of normality. Overall, the study underscores the significant bias present in OLS forward and reverse regressions due to measurement error and confirms that reverse IV esbmabon effecbvely corrects for this bias, ensuring consistency across both regression orientabons. 2.2. TheoreQcal Underpinnings of Reverse Regressions In describing the theorebcal underpinnings, the study adapts the notabons in Cochrane (2001) Hausman (2001) and Hanssen (2015). Suppose the variables (𝒚,𝑥) are jointly normally distributed. Consider the best predictor of 2 Measurement error in explanatory variables is a wellrecognized source of aCenuaQon bias, causing the ordinary least squares (OLS) esQmator of β to be biased towards zero (Greene, 2010; Hansen, 2015) Suppose the true model is: y = βx* + ε, but instead of observing x*, we observe x = x* + u, where u is classical measurement error. Estimating y = βx + ε' using OLS leads to bias: 𝛽!"# " → λβ where; 0 < λ < 1. This causes systematic underestimation of the effect of x* on y (Wooldridge, 2010; Hansen, 2015). Furthermore, regularizaQon methods such as 𝑦 given 𝒙 𝑦 = 𝒙!𝜷 + 𝑒 (1) 𝛽 = (𝔼(𝒙𝒙!))"𝟏/𝔼(𝒙𝑦) Given that the error 𝑒 is a linear transformation of the normal vector(𝑦,𝒙), it follows that ((𝑒,𝒙) is jointly normal. Additionally, given that 𝔼(𝒙𝑒)= 0 are jointly normal and uncorrelated, this means that they are also independent. Independence in this context implies that: 𝔼(𝑒|𝒙)= 𝔼(𝑒)= 0 and 𝔼(𝑒$|𝑥)= 𝔼(𝑒$)= 𝜎$ (2) Equation 2 denotes properties of a homoscedastic linear conditional expectation function, CEF (Hansen, 2015). Given that (𝑦,𝒙) are jointly normally distributed, they satisfy a normal linear CEF. 𝑦 = 𝒙!𝜷 + 𝑒'''''''''''''''''''''''''''''''''''''''''''''''''''''(3) Where 𝑒 ∼ 𝑁(0,𝜎$) is independent of 𝒙 The theoretical discussions in equations 1 and 2 represent the ‘traditional’ motivation for the linear CEF models. However, it is contended that this motivation has limited merit in econometric applications given that, on the whole, economic and finance data is usually non-normally distributed (Hausman, 2001; Hansen, 2015). Consider the classical linear model (CLM): 𝑌%= 𝛽𝑿%+ 𝜀% (4) Suppose 𝑿 is measured with error: 𝑿%= 𝑿%+ 𝑣% (5) Where 𝑣% is the classical measurement error or aGenuabon bias. This implies that the ordinary least squares (OLS) esbmator of 𝛽 is biased towards zero.2 In high-dimensional contexts, regularized variants of ordinary least squares, such as ridge regression, penalize the coefficients to draw them closer to zero, thereby decreasing variance. Although this process does not ridge regression inherently introduce a shrinkage bias by design, which pulls parameter esQmates closer to zero in order to reduce variance. The estimated β values are systematically pulled toward zero to minimize mean squared error (MSE). 𝛽$%&'( " = (X'X + λI)-1X'y as λ > 0, this introduces shrinkage and a bias toward zero (Hastie, Tibshirani, & Friedman, 2009). In environments characterized by high levels of noise or when omiCed variables are correlated with the included regressors, similar paCerns of bias may also arise (Einav & Levin, 2014). Leveraging Reverse Regressions for Bias Diagnosis in the Digital Economy Datasets 53 consbtute 'bias' in the tradibonal sense, the esbmated beta coefficients are deliberately biased toward zero to opbmize the mean squared error, MSE (Hausman, 2001). Given these sources of aGenuabon bias, reverse regression (RR) mibgates the aGenuabon bias by flipping the dependent and independent variables: 𝑿%= 𝛾&𝒀%+ 𝜂% (6) If 𝑋 is measured with error, the reverse regression slope 𝛾&/ can be compared to the inverse of 𝛽 (Cochrane ,2001). The slope/𝛾& will differ in magnitude and direcbon depending on the presence and nature of the measurement error or endogeneity. 3. METHODOLOGY 3.1. Data DescripQons The dataset employed in this research was synthetically created to mimic the typical features of digital economy data influenced by measurement errors. Specifically, the data represents a hypothetical online company focused on digital sales. The study envisions a scenario of a medium-sized enterprise business digital sales data associated with advertising expenditure reported via Google or Meta dashboards. The study evaluates the impact of digital advertising expenditure on digital sales. This dataset includes the true digital advertising spending unobserved in practice (following the model in Zeng et al.2008), the reported digital advertising spending with measurement error, the digital sales, measurement error (added to the true and reported digital advertising spending) and the random error (added to the digital sales. Utilizing conventional econometric simulation techniques3, the true independent variable; reported digital advertising expenditure, was sampled from a normal distribution. Subsequently, measurement errors were systematically incorporated to replicate the noisy reporting typical of digital datasets. The dependent variable, digital sales, was generated as a linear function of the true independent variable with the addition of random error. This methodology establishes a controlled setting to assess the efficacy of reverse regression as a diagnostic and corrective instrument in the presence of realistic data noises frequently 3 The study employed the Scenario Analysis to generate the hypotheQcal data following the Kydland & PrescoC (1982) model. The Scenario Analysis is supplemented by the staQc encountered in digital economy data analyses (Wei & Wright,2013; Ryan, & Yang, 2015). Figure 1 shows the scaGer plot for digital sales and digital (reported advert) adverbsing expenditure. The plot clearly shows there is a posibve correlabon between online reported adverbsing and digital sales. Figure 1. Scatter Plot Digital Advertising vs Digital Sales Source: Author’s elaboration on data The relatively positive correlation between online advertising expenditure and digital sales can be explained from the perspective that online adverts help make products more visible, raise consumer awareness, and attract more visitors to online stores where purchases happen. Several important factors explain this positive correlation. First, online adverts boost awareness of products and strengthen brand recognition, making consumers more likely to consider buying. When people see adverts on social media, search engines, or websites, they learn about product features, benefits, and special offers, which can spark interest and the desire to buy (Grewal et al., 2020). Second, targeted advertising allows companies to reach the right audiences, who are more likely to purchase. Digital platforms analyze user data and browsing habits to show adverts to individuals whose behavior indicates a higher chance of buying. This focused approach reduces wasted advertising expenditure and makes campaigns more effective (Lambrecht & Tucker, 2013). Third, interactive and personalized adverts microsimulaQon technique, given that the data generated are firm-level microeconomic data (Li & O’Donoghue, 2013). !" #" A"" A%" A&" A!" A#" A"" %"" D"" &"" ("" )*+*,-.///0-.12////*3////)4..-52 /)*+*,-.///S7/////08137*3+////*3/////)4..-52 Mulenga 54 help engage customers more deeply, which can lead to more conversions and potential increases in purchases. Features like clickable ads, product demos, and real-time feedback enable users to interact with products before deciding to buy, building trust and increasing the chances of purchase (Chaffey & Ellis-Chadwick, 2019). Lastly, the quick and easy nature of digital platforms enhances the impact of advertising. For instance, after seeing an ad, consumers can immediately click to view a product page, read reviews, and complete their purchase. This smooth process shortens the decision-making time and encourages more sales (Stephen, 2016). 3.2. SimulaQon Strategy Model 1: Direct OLS Regression 𝑌 = 𝛽'+ 𝛽&𝑿 + 𝑢////////////////////////////////////(7)/ Model 2: Reverse Regression 𝑿 = 𝛾'+ 𝛾&𝒀 + 𝜂/////////////////////////////////////(8) If 𝑋 is measured with error, the reverse regression slope, 𝛾&/ can be compared to the inverse of 𝛽 (Cochrane, 2001). That is, the bias indicator is estimated as: 𝐵𝑖𝑎𝑠 = H𝛽 I&H//|𝛾K&|/ (9) This ratio serves as a diagnostic measure of the degree and direction of potential bias or measurement error. A prori expectations: (i) When Bias = 0; forward and reverse regressions agree, however, (ii) When Bias > 0; the forward estimate is attenuated or biased towards zero and differs from the reciprocal of the regression slope. The measurement error is defined as: 𝑋%= 𝑋%+ 𝑣% ∗ , where 𝑣%~𝑁(0,𝜎$) . In summary, the simulation strategy involves running both direct and reverse regressions on simulated data, then comparing the estimated slopes and their statistical significance.4 4. RESULTS AND DISCUSSIONS Table 1 reports summary stabsbcs of the simulated data. Table 1. Summary Statistics DIG_SAL TRUE AD_SPEND REPORTED_AD_SPEN RANDOM_ERROR_SALE MEASUREMENT _ERROR Mean 275.1278 110.5114 110.2989 -1.150864 -0.212552 Median 274.2002 110.413 110.7415 -1.382851 -0.067832 Max. 454.4609 167.791 166.8657 105.8717 39.26238 Min. 121.3624 61.38099 54.00618 -110.651 -31.76704 Std. Dev. 48.21366 14.81024 17.81859 30.87028 10.05153 Sum 704327 282909.3 282365.2 -2946.212 -544.1321 Sum Sq. Dev. 5948540 561299.6 812487.8 2438661 258543.9 Obs. 2560 2560 2560 2560 2560 Notes: Dig_Sal is Digital Sales, Advert_Spend is digital AdverQsing Spending or expenditure, Reported_Ad_Spend is reported online AdverQsing Spending, Random_Error_Sale is Random error in Sales. Table 2 reports the correlabon relabonships among the variables in the sample. From Table 2, it seems both the true adverbsing spending (True_Ad_spending) and the reported spending (Reported_AD_Spending) indicates a relabvely strong posibve correlabon between digital expenditure and digital sales. This observabon is in tandem with the results obtained in the scaGer plot in Figure 1, where it is found that there is a posibve correlabon between online adverbsing expenditure and digital sales. The relabvely strong posibve correlabon shown in Table 2 can be explained by the fact that online adverts generally help make products more visible, raise consumer awareness, and aGract more visitors online or digital stores, which increases the probability of making purchases (Chaffey & Ellis-Chadwick, 2019; Grewal et al., 2020). 4 If suitable instrumental variables (IV) existed, the diagnosQcs would be compared with IV results obtained via instrumental variable regressions taking this format: 𝑌 = 𝛿)+ 𝛿*𝑋 (+ 𝜀, where 𝑋 ( is predicted from instrument 𝑍. Leveraging Reverse Regressions for Bias Diagnosis in the Digital Economy Datasets 55 Table 2. Correlation Matrix DIG_SA L AD_SPEN D REPORTED_AD_S PE RANDOM_ERROR_S A MEASUREMENT_ ERR DIG_SAL_ 1 TRUE_AD_SPEND 0.768 1 REPORTED_AD_SPEN D 0.636 0.826 1 RANDOM_ERROR_ 0.640 0.004 0.003 1 0.005 MEASUREMENT_ERR O -0.004 -0.009 0.556 0.005 1 Table 3 reports a summary of results obtained in the forward (direct) and reverse (backward) regressions employing equabons 7 and 8. The variables were in first difference format following the non-stabonarity test results of each variable in the unit root test diagnosbcs (not reported here). Table 3. Summary Results of Forward and Reverse Regressions Coeff Std.Error t-Stat P_value Mean Dependent Var Model 1: Forward Regressions (Method-Least Squares) Model 1A: Dependent variableSales with random error Independent VariableTrue-Ad-Spending 4.972** 0.204 24.387 0.00 273.976 Model 1B: Dependent variableSales with random error Independent VariableReported_Ad_Spending 3.174** 0.146 - 16.902 0.00 273.976 Model 2: Reverse Regressions Model 2A: Independent variableSales with random error Dependent Variable-True_Ad_Spending 2.706** 0.004 23.903 0.00 110.086 Model 2B: Dependent VariableReported_Ad_Spending 1.606** 0.007 16.262 0.00 110.086 Source: Author’s elaboraQon on simulated data. ** denotes staQsQcal significance at 5%. Results in Table 3 indicate that both true and reported digital adverbsing spending have posibve, stabsbcally significant effects on digital sales in the forward regressions (model 1). Specifically, ceteris paribus, an increase of 1 unit in true digital/online adverbsing expenditure causes an increase in digital sales of 4.97 %, and a unit increase in reported online adverbsing expenditure increases digital sales by 2.47%. Similarly, in model 2, the reverse regressions indicate that both the true and reported digital expenditures have a posibve and significant effect on digital sales. Specifically, holding other factors constant, the true and reported digital expenditures increase digital sales by 2.71% and 3.12%, respecbvely. The posibve significant effect of both the true and reported digital expenditures on digital sales can be explained in part by the fact that online adverts help make products more visible, raise consumer awareness, and aGract more visitors to online stores where purchases happen. Online adverts boost awareness of products and strengthen brand recognibon, making consumers more likely to consider buying. When people see adverts on social media, search engines, or websites, they learn about product features, benefits, and special offers, which can spark interest and the desire to buy (Grewal et al., 2020). Features like clickable ads, product demos, and real-bme feedback enable users to interact with products before deciding to buy, building trust and increasing the chances of purchase (Chaffey & Ellis-Chadwick, 2019). 4.1. Determining the AYenuaQon Bias (esQmaQng the bias indicator) This secbon employs equabon 9 to esbmate the degree and direction of bias (or measurement error. That is; 𝐵𝑖𝑎𝑠 = H𝛽 I&H//|𝛾K&|/