{ "id": "ep4wq7b1a8", "slug": "distributed-lag-model", "title": "Distributed lag model", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.regression-analysis" }, { "id": "physical.math.statistics.stats_statistical_inference.regression-analysis.time-series-regression", "label": "Time series regression", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.regression-analysis.time-series-regression" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Non-geographic · 1800 to 1945: Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.regression-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics" }, { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.regression-analysis" } ] }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Non-geographic · 1946 to 2000: Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis" } ] } ], "excerpt": "A distributed lag model is a time series regression that estimates a predictor's effect on an outcome as spread over several past values, with coefficients called lag weights.", "snippet": "A distributed lag model is a time series regression that estimates a predictor's effect on an outcome as spread over several past values, with coefficients called lag weights.", "node": "physical.math.statistics.stats_statistical_inference.regression-analysis.time-series-regression", "markdown": "# Distributed lag model\n\nA distributed lag model is a regression that estimates the effect of a predictor on an outcome as spread over several past values of the predictor rather than concentrated in a single period. In its finite form the coefficients \\( \\beta_i \\) are called lag weights and together form the lag distribution, the pattern of how \\( X \\) affects \\( y \\) over time.[1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf) [Software documentation](https://www.edgepedia.org/software-documentation) reserves the term for models in which the predictor differs from the outcome; a model of a variable on its own past values is an autoregression.[2](https://www.estima.com/webhelp/topics/reg-distributedlags.html)\n\n| Key fact | Statement |\n| --- | --- |\n| Regression form | \\( y_t = \\alpha + \\sum_{i=0}^{k} \\beta_i X_{t-i} + u_t \\), estimated by OLS, but with large \\( k \\) the lagged explanators are strongly collinear and restrictions on the coefficients are typically needed[3](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_2010_C_DistributedLags.pdf) |\n| Interpretation | \\( \\beta_h \\) is the \\( h \\)-period dynamic multiplier and \\( \\beta_0 \\) is the impact (contemporaneous) effect[4](http://www.econometrics-with-r.org/15.3-dynamic-multipliers-and-cumulative-dynamic-multipliers.html) |\n| Cumulative effect | The cumulative dynamic multipliers are the coefficients of a modified regression in differences of \\( X \\), estimable directly by OLS with HAC standard errors[4](http://www.econometrics-with-r.org/15.3-dynamic-multipliers-and-cumulative-dynamic-multipliers.html) |\n| Main weakness | Multicollinearity increases with lag order because multiple lags of the same series enter the model[5](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0228812) |\n| Classic restriction | The geometric (Koyck) lag sets the coefficients to \\( \\beta_0\\lambda^{j} \\), giving a long-run effect of \\( \\beta_0/(1-\\lambda) \\)[6](https://web.pdx.edu/~crkl/ec571/lecture10.htm) |\n| Modern form | Distributed lag non-linear models combine a predictor basis and a lag basis into a cross-basis[7](https://onlinelibrary.wiley.com/doi/10.1002/sim.3940) |\n\n## How it works\n\nEach lag weight \\( \\beta_i \\) measures the effect of \\( X_{t-i} \\) on \\( y_t \\), holding earlier lags fixed. The \\( h \\)-period dynamic multiplier is \\( \\beta_h \\), and the \\( h \\)-period cumulative dynamic multiplier is the running sum of the dynamic multipliers from lag 0 through lag \\( h \\).[4](http://www.econometrics-with-r.org/15.3-dynamic-multipliers-and-cumulative-dynamic-multipliers.html) Cumulative multipliers can be read directly from a reparameterized regression \\( Y_t = \\delta_0 + \\delta_1 \\Delta X_t + \\cdots + \\delta_r \\Delta X_{t-r+1} + \\delta_{r+1} X_{t-r} + u_t \\), whose coefficients \\( \\delta_1, \\ldots, \\delta_{r+1} \\) are the cumulative multipliers.[4](http://www.econometrics-with-r.org/15.3-dynamic-multipliers-and-cumulative-dynamic-multipliers.html) With two lags, the long-run propensity is \\( \\mathrm{LRP} \\equiv \\delta_3 \\), the coefficient on \\( X_{t-2} \\); multicollinearity can leave the individual short-run multipliers imprecise while the LRP remains well estimated.[8](https://www.fsb.miamioh.edu/lij14/311_2014_0428.pdf) Under the geometric scheme, the mean lag is \\( \\lambda/(1-\\lambda) \\) and the median lag, defined as the smallest integer \\( m \\geq 0 \\) with \\( 1 - \\lambda^{m+1} \\geq 0.5 \\), is approximated continuously by \\( \\ln(0.5)/\\ln(\\lambda) \\).[6](https://web.pdx.edu/~crkl/ec571/lecture10.htm)\n\n## How it is done\n\nThere is no single right way to identify the lag length.[1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf) Common practice adds lags until the residuals appear to be white noise, tested with a Breusch-Godfrey LM test or a Box-Ljung Q test, or uses trailing-lag significance tests.[1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf) Each one-period increase in the lag costs two degrees of freedom, one for the extra coefficient and one for the lost presample observation, leaving \\( T - 2q - 2 \\) degrees of freedom with one regressor.[1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf) A lag length below the true length gives biased estimators (omitted variables); a length above it gives inefficient estimators (irrelevant variables).[9](https://www.aadecon.com/classes/econ5360/notes2.pdf) Akaike's criterion, \\( \\mathrm{AIC}(n) = \\ln \\tilde{\\sigma}^2_n + 2n/T \\), is a popular selector, and a two-step procedure estimates the lag length first and then the polynomial degree.[9](https://www.aadecon.com/classes/econ5360/notes2.pdf) • [1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf) [Information](https://www.edgepedia.org/information) criteria must compare regressions on samples of identical length.[1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf) When \\( X \\) is highly serially correlated, information criteria often simply pick the maximum allowed lag.[2](https://www.estima.com/webhelp/topics/reg-distributedlags.html) [Estimation](https://www.edgepedia.org/estimation) is by OLS with HAC standard errors, because the errors of a distributed lag model are generally serially correlated.[10](https://www.econometrics-with-r.org/15.5-estimation-of-dynamic-causal-effects-with-strictly-exogeneous-regressors.html)\n\n## Origin\n\nDistributed lag analysis grew out of econometric work on investment and demand. The geometric lag scheme carries L. M. Koyck's name through his investment analysis, documented in a 1955 The Economic Journal record of Distributed Lags and Investment Analysis.[11](https://doi.org/10.2307/2227337) L. R. Klein's 1958 [Econometrica](https://www.edgepedia.org/econometrica) paper \"The Estimation of Distributed Lags\" is an early methodological treatment.[12](https://doi.org/10.2307/1907516) The polynomial distributed lag takes its name from Shirley Almon's 1965 Econometrica paper on the lag between capital appropriations and expenditures.[13](https://doi.org/10.2307/1911894) Phoebus J. Dhrymes, [Lawrence R. Klein](https://www.edgepedia.org/lawrence-r-klein), and Kenneth Steiglitz gave a maximum-likelihood formulation for general rational lag structures in a 1970 International Economic Review paper, building on an engineering idea of Steiglitz and McBride.[14](https://doi.org/10.2307/2525666) Softer restrictions followed: a smoothness-prior estimator[15](https://doi.org/10.2307/1914096) and ridge estimators.[16](https://doi.org/10.3386/w0069) Takeshi Amemiya and Kimio Morimune (1974, The Review of Economics and [Statistics](https://www.edgepedia.org/statistics)) studied the optimal polynomial order.[17](https://doi.org/10.2307/1923977) In epidemiology, a polynomial distributed lag for air pollution and daily deaths appeared in 2000 (Joel Schwartz, Epidemiology),[18](https://doi.org/10.1097/00001648-200005000-00016) followed by generalized additive distributed lag models for mortality displacement[19](https://doi.org/10.1093/biostatistics/1.3.279) and temperature-mortality models that generalized toward non-linear lag structures.[20](https://doi.org/10.1097/01.ede.0000239732.50999.8f) The DLNM framework and the R package dlnm appear in A. Gasparrini, B. Armstrong, and M. G. Kenward's 2010 Statistics in Medicine paper and Gasparrini's 2011 Journal of Statistical Software paper.[7](https://onlinelibrary.wiley.com/doi/10.1002/sim.3940) • [21](https://doi.org/10.18637/jss.v043.i08)\n\n## Variants\n\n**Geometric (Koyck) lag.** Coefficients \\( \\beta_0\\lambda^{j} \\) with \\( 0 < \\lambda < 1 \\) (equivalently, normalized weights \\( (1-\\lambda)\\lambda^{j} \\) scaled by the long-run effect) reduce the model to \\( y_t = (1-\\lambda)\\alpha + \\lambda y_{t-1} + \\beta(1-\\lambda)X_t + (u_t - \\lambda u_{t-1}) \\), a lagged-dependent-variable form with autocorrelated moving-average errors.[3](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_2010_C_DistributedLags.pdf) The long-run effect is \\( \\beta_0/(1-\\lambda) \\).[6](https://web.pdx.edu/~crkl/ec571/lecture10.htm)\n\n**Polynomial (Almon) lag.** The weights are constrained to a polynomial, \\( \\beta_i = a_0 + a_1 i + \\cdots + a_r i^{r} \\) with \\( r < k \\), reducing the number of estimated parameters;[3](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_2010_C_DistributedLags.pdf) the estimator is restricted least squares subject to linear homogeneous restrictions, end-point (tie-down) restrictions are allowed, and the restrictions can be tested as linear restrictions on unrestricted estimates.[9](https://www.aadecon.com/classes/econ5360/notes2.pdf) • [6](https://web.pdx.edu/~crkl/ec571/lecture10.htm) • [1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf)\n\n**DLNMs and penalized models.** The DLNM cross-basis combines two sets of basis functions (splines, polynomials, strata, thresholds) for predictor and lag through a tensor product, estimated with standard regression commands.[7](https://onlinelibrary.wiley.com/doi/10.1002/sim.3940) • [21](https://doi.org/10.18637/jss.v043.i08) A penalized extension fits DLNMs as penalized splines within GAMs, with built-in model selection and improved inferential properties over the unpenalized version.[22](https://doi.org/10.1111/biom.12645) Bayesian treatments place priors on the lag shape;[23](https://doi.org/10.1002/asmb.628) in simulations, shrinkage methods (generalized ridge, hierarchical Bayes) outperform both constrained and unconstrained DLMs under misspecification.[24](https://escholarship.org/content/qt8g0602dj/qt8g0602dj.pdf) The adaptive cumulative exposure [DLNM](https://www.edgepedia.org/distributed-lag-non-linear-model) defines \\( E(t) = \\int_{0}^{L} w(l) X(t-l) \\, dl \\) with an unknown smooth weight \\( w \\); fixing \\( w \\) gives a GAM and fixing the exposure function to linear gives a classic DLM.[25](https://arxiv.org/html/2505.15759v1) Distributed lag quantile regression extends the framework to quantiles of time-dependent exposure mixtures.[26](https://doi.org/10.1111/biom.13702)\n\n## Applications\n\n**Econometrics.** Classic uses include investment and demand analysis, where the lag distribution summarizes how spending or demand responds over time.[11](https://doi.org/10.2307/2227337) • [12](https://doi.org/10.2307/1907516) In macroeconomics the ARDL model, which adds an autoregressive component to the distributed lag, delivers dynamic responses by polynomial division \\( \\beta(L)/\\alpha(L) \\) and a long-run response evaluated at \\( L = 1 \\);[2](https://www.estima.com/webhelp/topics/reg-distributedlags.html) it provides consistent long-run estimates even when explanatory variables are weakly endogenous.[27](https://ghe.yonsei.ac.kr/yeri/publications/report.do?articleNo=120226&attachNo=101995&mode=download)\n\n**Environmental epidemiology.** DLMs and DLNMs are used for air pollution and temperature-mortality associations. The dlnmTS vignette illustrates a Chicago NMMAPS analysis with a temperature DLNM, finding that cold is associated with longer-lasting mortality risk than heat, which shows a \"protective\" effect at lag 0.[28](https://gasparrini.r-universe.dev/dlnm/doc/dlnmTS.pdf) Attributable risk measures (backward and forward attributable fractions and numbers) extend the framework to burden estimation.[29](https://doi.org/10.1186/1471-2288-14-55)\n\n**Software.** In R, dlnm provides crossbasis(), crosspred() (lag-specific and overall cumulative effects with 95% confidence intervals by default), crossreduce(), and penalization via cbPen or mgcv.[30](https://cran.r-project.org/web/packages/dlnm/refman/dlnm.html) dLagM estimates the Koyck model by instrumental variables and implements the [ARDL bounds test](https://www.edgepedia.org/ardl-bounds-test).[5](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0228812) Stata's ardl selects lag orders by AIC or BIC and offers the bounds test as postestimation.[31](https://journals.sagepub.com/doi/10.1177/1536867X231212434) RATS documentation lists five estimation approaches: unrestricted long lags, data-determined lag length, hard shape restrictions (Almon polynomials, splines), soft restrictions (smoothness priors), and ARDL models.[2](https://www.estima.com/webhelp/topics/reg-distributedlags.html)\n\n## Limitations and alternatives\n\n**Multicollinearity.** Because lagged values of a serially correlated series are highly correlated, individual coefficients in an unrestricted distributed lag are poorly determined; in one documented example only lags 0 and 24 are individually significant, so summary measures such as the sum of lag coefficients are of primary interest.[2](https://www.estima.com/webhelp/topics/reg-distributedlags.html) Imposing a flat lag distribution reduces the model to regressing \\( y \\) on the summed variable \\( Z = X_t + X_{t-1} + X_{t-2} \\).[8](https://www.fsb.miamioh.edu/lij14/311_2014_0428.pdf)\n\n**Specification and inference.** Beyond the biased-or-inefficient lag-length tradeoff, [Monte Carlo](https://www.edgepedia.org/monte-carlo) work finds the Durbin-Watson statistic of little use for assessing lag length, and omitting a non-lagged variable can make even spurious lag weights appear significant.[32](https://www.nber.org/system/files/chapters/c9838/c9838.pdf) The Koyck transformation creates a composite error \\( \\upsilon_i = \\varepsilon_i - \\lambda \\varepsilon_{i-1} \\) correlated with the lagged dependent variable, so OLS is inconsistent; dLagM therefore uses instrumental variables, with the Wu-[Hausman test](https://www.edgepedia.org/hausman-test) for endogeneity applied cautiously in small samples.[6](https://web.pdx.edu/~crkl/ec571/lecture10.htm) • [5](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0228812) With a Koyck model and AR(1) errors, the Hildreth-Lu search is valid while Prais-Winsten and Cochrane-Orcutt are not.[1](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf)\n\n**Alternatives.** Moving-average models approximate lag patterns acceptably only for short, correctly specified lag periods; used for long or complex lag patterns, or with the wrong interval, they produce substantial biases, whereas DLMs and DLNMs show no or low bias with close-to-nominal confidence intervals even for long lags under strong seasonal trends.[33](https://pmc.ncbi.nlm.nih.gov/articles/PMC5388182/) ARDL models tolerate weakly endogenous regressors and support a bounds test valid for mixtures of I(0) and I(1) series, with AIC selecting less parsimonious models than BIC when serial correlation is a concern;[31](https://journals.sagepub.com/doi/10.1177/1536867X231212434) when the outcome also determines the explanatory variables' long-run equilibrium, a VAR or VECM is preferable.[31](https://journals.sagepub.com/doi/10.1177/1536867X231212434)\n\n## References\n\n1. [Time-Series Analysis chapter on distributed-lag models (Parker, Reed College)](https://www.reed.edu/economics/parker/s14/312/tschapters/S13_Ch_3.pdf)\n2. [Distributed Lags (RATS software documentation)](https://www.estima.com/webhelp/topics/reg-distributedlags.html)\n3. [Distributed lag models (lecture notes, Jean-Marie Dufour, McGill University)](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_2010_C_DistributedLags.pdf)\n4. [15.3 Dynamic Multipliers and Cumulative Dynamic Multipliers (Introduction to Econometrics with R)](http://www.econometrics-with-r.org/15.3-dynamic-multipliers-and-cumulative-dynamic-multipliers.html)\n5. [dLagM: An R package for distributed lag models and ARDL bounds testing (PLOS One)](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0228812)\n6. [EC 570/571 Topic 10: Distributed Lag Models (Portland State University lecture notes)](https://web.pdx.edu/~crkl/ec571/lecture10.htm)\n7. [Distributed lag non-linear models (Gasparrini, Armstrong, Kenward, 2010, Statistics in Medicine)](https://onlinelibrary.wiley.com/doi/10.1002/sim.3940)\n8. [Time series lecture notes (Miami University)](https://www.fsb.miamioh.edu/lij14/311_2014_0428.pdf)\n9. [Distributed Lag Models (Berry & Wei, University of Wyoming, 2013)](https://www.aadecon.com/classes/econ5360/notes2.pdf)\n10. [Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors (Introduction to Econometrics with R)](https://www.econometrics-with-r.org/15.5-estimation-of-dynamic-causal-effects-with-strictly-exogeneous-regressors.html)\n11. [L. R. Klein, L. M. Koyck, H. Goris (1955). Distributed Lags and Investment Analysis.. The Economic Journal.](https://doi.org/10.2307/2227337)\n12. [L. R. Klein (1958). The Estimation of Distributed Lags. Econometrica.](https://doi.org/10.2307/1907516)\n13. [Shirley Almon (1965). The Distributed Lag Between Capital Appropriations and Expenditures. Econometrica.](https://doi.org/10.2307/1911894)\n14. [Phoebus J. Dhrymes, Lawrence R. Klein, Kenneth Steiglitz (1970). Estimation of Distributed Lags. International Economic Review.](https://doi.org/10.2307/2525666)\n15. [Robert J. Shiller (1973). A Distributed Lag Estimator Derived from Smoothness Priors. Econometrica.](https://doi.org/10.2307/1914096)\n16. [G.S. Maddala (1974). Ridge Estimators for Distributed Lag Models. National Bureau of Economic Research.](https://doi.org/10.3386/w0069)\n17. [Takeshi Amemiya, Kimio Morimune (1974). Selecting the Optimal Order of Polynomial in the Almon Distributed Lag. The Review of Economics and Statistics.](https://doi.org/10.2307/1923977)\n18. [Joel Schwartz (2000). The Distributed Lag between Air Pollution and Daily Deaths. Epidemiology.](https://doi.org/10.1097/00001648-200005000-00016)\n19. [A. Zanobetti (2000). Generalized additive distributed lag models: quantifying mortality displacement. Biostatistics.](https://doi.org/10.1093/biostatistics/1.3.279)\n20. [Ben Armstrong (2006). Models for the Relationship Between Ambient Temperature and Daily Mortality. Epidemiology.](https://doi.org/10.1097/01.ede.0000239732.50999.8f)\n21. [Antonio Gasparrini (2011). Distributed Lag Linear and Non-Linear Models in R : The Package dlnm. Journal of Statistical Software.](https://doi.org/10.18637/jss.v043.i08)\n22. [Antonio Gasparrini and colleagues (2017). A Penalized Framework for Distributed Lag Non-Linear Models. Biometrics.](https://doi.org/10.1111/biom.12645)\n23. [Romy R. Ravines, Alexandra M. Schmidt, Helio S. Migon (2006). Revisiting distributed lag models through a Bayesian perspective. Applied Stochastic Models in Business and Industry.](https://doi.org/10.1002/asmb.628)\n24. [Shrinkage estimation of distributed lag models (UC eScholarship)](https://escholarship.org/content/qt8g0602dj/qt8g0602dj.pdf)\n25. [Estimating Associations Between Cumulative Exposure and Health via Generalized Distributed Lag Non-Linear Models using Penalized Splines (2025 preprint)](https://arxiv.org/html/2505.15759v1)\n26. [Yuyan Wang and colleagues (2022). Semiparametric Distributed Lag Quantile Regression for Modeling Time-Dependent Exposure Mixtures. Biometrics.](https://doi.org/10.1111/biom.13702)\n27. [Review of the ARDL model literature (Yonsei Economic Research Institute)](https://ghe.yonsei.ac.kr/yeri/publications/report.do?articleNo=120226&attachNo=101995&mode=download)\n28. [dlnmTS vignette: Distributed lag linear and non-linear models for time series](https://gasparrini.r-universe.dev/dlnm/doc/dlnmTS.pdf)\n29. [Antonio Gasparrini, Michela Leone (2014). Attributable risk from distributed lag models. BMC Medical Research Methodology.](https://doi.org/10.1186/1471-2288-14-55)\n30. [dlnm package reference manual (CRAN)](https://cran.r-project.org/web/packages/dlnm/refman/dlnm.html)\n31. [ardl: Estimating autoregressive distributed lag and equilibrium correction models (Stata Journal)](https://journals.sagepub.com/doi/10.1177/1536867X231212434)\n32. [A Monte Carlo Study of Complex Finite Distributed Lag Structures](https://www.nber.org/system/files/chapters/c9838/c9838.pdf)\n33. [Modelling lagged associations in environmental time series data: a simulation study](https://pmc.ncbi.nlm.nih.gov/articles/PMC5388182/)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Regression analysis › Time series regression*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://web.pdx.edu/~crkl/ec571/lecture10.htm" ], "url": "https://www.edgepedia.org/distributed-lag-model", "markdown_url": "https://www.edgepedia.org/distributed-lag-model.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Distributed lag model\", Edgepedia (EdgeChat), https://www.edgepedia.org/distributed-lag-model. Edgepedia Community License 1.0.", "credit_md": "\"[Distributed lag model](https://www.edgepedia.org/distributed-lag-model)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/distributed-lag-model](https://www.edgepedia.org/distributed-lag-model). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Distributed lag model\", Edgepedia (EdgeChat), https://www.edgepedia.org/distributed-lag-model. Edgepedia Community License 1.0.", "speakable": "A distributed lag model is a time series regression that estimates a predictor's effect on an outcome as spread over several past values, with coefficients called lag weights." }