Understanding how households escape poverty depends on understanding how they accumulate assets over time. Therefore, identifying the degree of linearity in household asset dynamics, and specifically any potential asset poverty thresholds, is of fundamental interest to the design of poverty reduction policies. If household asset holdings converged unconditionally to a single long run equilibrium, then all poor could be expected to escape poverty over time. In contrast, if there are critical asset thresholds that trap households below the poverty line, then households would need specific assistance to escape poverty. Similarly, the presence of asset poverty thresholds would mean that short term asset shocks could lead to long term destitution, thus highlighting the need for social safety nets. In addition to the direct policy relevance, identifying household asset dynamics and potential asset thresholds presents an interesting methodological challenge to researchers. Potential asset poverty thresholds can only be identified in a framework that allows multiple dynamic equilibria. Any unstable equilibrium points would indicate a potential poverty threshold, above which households are expected to accumulate further and below which households are on a trajectory that makes them poorer over time. The key empirical issue addressed in the paper is whether such threshold points exist in Pakistan and Ethiopia and, if so, where they are located. Methodologically, the paper explores what econometric technique is best suited for this type of analysis. The paper contributes to the small current literature on modeling nonlinear household welfare dynamics in three ways. First, it compares previously used techniques for identifying asset poverty traps by applying them to the same dataset, and examines whether, and how, the choice of estimation technique affects the result. Second, it explores whether other estimation techniques may be more suitable to locate poverty thresholds. Third, it adds the first study for a South Asian country and makes a comparison with Ethiopia. Household assets are combined into a single asset index using two techniques: factor analysis and regression. These indices are used to estimate asset dynamics and locate dynamic asset equilibria, first by nonparametric methods including LOWESS, kernel weighted local regression and spline smoothers, and then by global polynomial parametric techniques. To combine the advantages of nonparametric and parametric techniques - a flexible functional form and the ability to control for covariates, respectively - the paper adapts a mixed model representation of a penalized spline to estimate asset dynamics through a semiparametric partially linear model. This paper identifies a single dynamic asset equilibrium with a slightly concave dynamic asset accumulation path in each country. There is no evidence for multiple dynamic equilibria. This result is robust across econometric methods and across different ways of constructing the asset index. The concave accumulation path means that poorer households recover more slowly from asset shocks. Concavity also implies that greater initial equality of assets would lead to higher growth. Moreover, the dynamic asset equilibria are very low. In Pakistan it is below the average asset holdings of the poor households in the sample. In Ethiopia, the equilibrium is barely above the very low mean. This, together with the slow speed of asset accumulation for the poorest households, suggests that convergence towards the long run equilibrium may be slow and insufficient for rural households in Pakistan and Ethiopia to escape poverty.